2-D Euler Shape Design on Nonregular Flows Using Adjoint Rankine-Hugoniot Relations - Baeza, Antonio; Castro, Carlos; Palacios, Francisco; Zuazua, Enrique (Aiaa Journal, 2009)
Optimal aerodynamic shape design aims to find the minimum of a functional that describes an aerodynamic property, by controlling the partial differential equation modeling the dynamics of the flow that surrounds an aircraft, by using surface deformation techniques. As a solution to the enormous computational resources required for classical shape optimization of functionals of aerodynamic interest, probably the best strategy is to apply methods inspired in control theory. One of the key ingredients relies on the usage of the adjoint methodology to simplify the computation of gradients. In this paper we restrict our attention to optimal shape design in two-dimensional systems governed by the steady Euler equations for flows whose steady-state solutions present discontinuities in the flow variables (an isolated shock wave). We first review some facts on control theory applied to optimal shape design and recall the 2-D Euler equations (including the Rankine-Hugoniot conditions). We then study the adjoint formulation, providing a detailed exposition of how the derivatives of functionals of aeronautical interest may be obtained when a discontinuity appears. Further on, adjoint equations will be discretized and analyzed and some novel numerical experiments with adjoint Rankine-Hugoniot relations will be shown. Finally, we expose some conclusions about the viability of a rigorous approach to the continuous Enter adjoint system with discontinuities in the flow variables.
CRIS-Team WoS Importer:2019-12-13

Actuator design for parabolic distributed parameter systems with the moment method - Privat, Yannick; Trelat, Emmanuel; Zuazua, Enrique (SIAM Journal on Control and Optimization, 2017)
In this paper, we model and solve the problem of designing in an optimal way actuators for parabolic partial differential equations settled on a bounded open connected subset of Rn. We optimize not only the location but also the shape of actuators, by finding what is the optimal distribution of actuators in , over all possible such distributions of a given measure. Using the moment method, we formulate a spectral optimal design problem, which consists of maximizing a criterion corresponding to an average over random initial data of the largest L2-energy of controllers. Since we choose the moment method to control the PDE, our study mainly covers one-dimensional parabolic operators, but we also provide several examples in higher dimensions. We consider two types of controllers: Either internal controls, modeled by characteristic functions, or lumped controls, that are tensorized functions in time and space. Under appropriate spectral assumptions, we prove existence and uniqueness of an optimal actuator distribution, and we provide a simple computation procedure. Numerical simulations illustrate our results.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Heat equation; Lumped control; Moment method; Null controllability; Parabolic systems; Shape optimization

Addendum: Local Elliptic Regularity for the Dirichlet Fractional Laplacian - Biccari, Umberto; Warma, Mahamadi; Zuazua, Enrique (Advanced Nonlinear Studies, 2017)
In [1], for 1 < p < 1
CRIS-Team Scopus Importer:2021-06-15
Keywords: Dirichlet Boundary Condition; Fractional Laplacian; Local Regularity; Weak Solutions

Adjoint computational methods for 2D inverse design of linear transport equations on unstructured grids - Morales-Hernandez, M.; Zuazua, Enrique (Computational and Applied Mathematics, 2019)
We address the problem of inverse design of linear hyperbolic transport equations in 2D heterogeneous media. We develop numerical algorithms based on gradient-adjoint methodologies on unstructured grids. While the flow equation is compulsorily solved by means of a second order upwind scheme so to guarantee sufficient accuracy, the necessity of using the same order of approximation when solving the sensitivity or adjoint equation is examined. Two test cases, including Doswell frontogenesis, are analysed. We show the convenience of using a low order method for the adjoint resolution, both in terms of accuracy and efficiency. An analytical explanation for this fact is also provided in the sense that, when employing higher order schemes for the adjoint problem, spurious high frequency numerical components slow down the convergence process.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Adjoint; First and second order schemes; Gradient descent method; Inverse design; Linear transport; Sensitivity

A framework for randomized time-splitting in linear-quadratic optimal control - Veldman, Daniel; Zuazua Iriondo, Enrique (Numerische Mathematik, 2022)
Inspired by the successes of stochastic algorithms in the training of deep neural networks and the simulation of interacting particle systems, we propose and analyze a framework for randomized time-splitting in linear-quadratic optimal control. In our proposed framework, the linear dynamics of the original problem is replaced by a randomized dynamics. To obtain the randomized dynamics, the system matrix is split into simpler submatrices and the time interval of interest is split into subintervals. The randomized dynamics is then found by selecting randomly one or more submatrices in each subinterval. We show that the dynamics, the minimal values of the cost functional, and the optimal control obtained with the proposed randomized time-splitting method converge in expectation to their analogues in the original problem when the time grid is refined. The derived convergence rates are validated in several numerical experiments. Our numerical results also indicate that the proposed method can lead to a reduction in computational cost for the simulation and optimal control of large-scale linear dynamical systems.
Keywords: Random Batch Method; Operator Splitting; Optimal Control; Model Predictive Control

A generic uniqueness result for the stokes system and its control theoretical consequences - Lions, Jacques Louis; Zuazua, Enrique (Partial Differential Equations and Applications: Collected Papers in Honor of Carlo Pucci, 2017)
We consider the Stokes system in a three-dimensional cylinder Ω = G × (0, L) of R3, G being a bounded smooth domain of R2. We study the following uniqueness property: If u is a solution of the Stokes system in Ω × (0, T) with Dirichlet boundary conditions, T being a positive time, and its third component vanishes, i.e. u 3 = 0, then can we ensure that u = 0? We prove that this property does hold for “almost every” crosssection G. By using the Fourier expansion of solutions the problem is reduced to show that, generically with respect to the cross-section G, there is no eigenfunction of the Stokes system with third component identically zero. We also show how this uniqueness result can be applied to obtain approximate controllability properties of the Stokes system with scalar controls oriented in the direction (0,0,1) of R3. Actually, it was while working on the approximate controllability problem that we were led to the problem of generic uniqueness studied in the present paper. We also prove that the results above fail when G is a ball of R2.
CRIS-Team Scopus Importer:2021-06-15

Allee optimal control of a system in ecology - Trelat, Emmanuel; Zhu, Jiamin; Zuazua, Enrique (Mathematical Models & Methods in Applied Sciences, 2018)
The Allee threshold of an ecological system distinguishes the sign of population growth either towards extinction or to carrying capacity. In practice, human interventions can tune the Allee threshold for instance thanks to the sterile male technique and the mating disruption. In this paper, we address various control problems for a system described by a diffusion-reaction equation regulating the Allee threshold, viewed as a real parameter determining the unstable equilibrium of the bistable nonlinear reaction term. We prove that this system is the mean field limit of an interacting system of particles in which the individual behaviour is driven by stochastic laws. Numerical simulations of the stochastic process show that the propagation of population is governed by travelling wave solutions of the macroscopic reaction-diffusion system, which model the fact that solutions, in bounded space domains, reach asymptotically an equilibrium configuration. An optimal control problem for the macroscopic model is then introduced with the objective of steering the system to a target travelling wave. Using well-known analytical results and stability properties of travelling waves, we show that well-chosen piecewise constant controls allow to reach the target approximately in sufficiently long time. We then develop a direct computational method and show its efficiency for computing such controls in various numerical simulations. Finally, we show the effectiveness of the obtained macroscopic optimal controls in the microscopic system of interacting particles and we discuss their advantage when addressing situations that are out of reach for the analytical methods. We conclude the paper with some open problems and directions for future research.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Allee effect; Diffusion-reaction equation; Direct computational method; Ecological system; Interacting particle system; Optimal control; Piecewise control strategy; Stochastic process; Travelling wave

An alternating descent method for the optimal control of the inviscid burgers equation in the presence of shocks - Castro, Carlos; Palacios, Francisco; Zuazua, Enrique (Mathematical Models & Methods in Applied Sciences, 2008)
We introduce a new optimization strategy to compute numerical approximations of minimizers for optimal control problems governed by scalar conservation laws in the presence of shocks. We focus on the 1 - d inviscid Burgers equation. We first prove the existence of minimizers and, by a Gamma convergence argument, the convergence of discrete minima obtained by means of numerical approximation schemes satisfying the so-called one-sided Lipschitz condition (OSLC). Then we address the problem of developing efficient descent algorithms. We first consider and compare the existing two possible approaches: the so-called discrete approach, based on a direct computation of gradients in the discrete problem and the so-called continuous one, where the discrete descent direction is obtained as a discrete copy of the continuous one. When optimal solutions have shock discontinuities, both approaches produce highly oscillating minimizing sequences and the effective descent rate is very weak. As a solution we propose a new method, that we shall call alternating descent method, that uses the recent developments of generalized tangent vectors and the linearization around discontinuous solutions. This method distinguishes and alternates the descent directions that move the shock and those that perturb the profile of the solution away of it producing very efficient and fast descent algorithms.
CRIS-Team WoS Importer:2019-12-13

ANALYSIS AND NUMERICAL SOLVABILITY OF BACKWARD-FORWARD CONSERVATION LAWS - Liard, Thibault; Zuazua Iriondo, Enrique (SIAM Journal on Mathematical Analysis, 2023)
In this paper, we study the problem of initial data identification for weak-entropy solutions of the one-dimensional Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the time-irreversibility of the Burgers equation, some target functions are unattainable from solutions of this equation, making the identification problem under consideration ill-posed. To get around this issue, we introduce a nonsmooth optimization problem, which consists in minimizing the difference between the predictions of the Burgers equation and the observations of the system at a final time in L2(ℝ) norm. Here, we characterize the set of minimizers of the aforementioned nonsmooth optimization problem. One of the minimizers is the backward entropy solution, constructed using a backward-forward method. Some simulations are given using a wave-front tracking algorithm.
CRIS-Team Scopus Importer:2023-06-30
Keywords: backward-forward method; conservation laws; identification problems; nonsmooth optimization problem; wave-front tracking algorithm; weak-entropy solutions

Analysis and numerical solvability of backward-forward conservation laws - Liard, Thibault; Zuazua Iriondo, Enrique (2024)
In this paper, we study the problem of initial data identification for weak-entropy solutions of the one-dimensional Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the time-irreversibility of the Burgers equation, some target functions are unattainable from solutions of this equation, making the identification problem under consideration ill-posed. To get around this issue, we introduce a non-smooth optimization problem, which consists in minimizing the difference between the predictions of the Burgers equation and the observations of the system at a final time in L2(R) norm. Here, we characterize the set of minimizers of the aforementioned non-smooth optimization problem. One of the minimizers is the backward entropy solution, constructed using a backward forward method. Some simulations are given using a wave-front tracking algorithm. <br
https://cris.fau.de/converis/publicweb/Publication/269652687
Keywords: Backward-forward method; Identification problems; Conservation Laws; Weak-entropy solutions; Non-smooth optimization problem; Wave-front tracking algorithm

An energy-based vibration model for beam bridges with multiple constraints - Huang, Shiping; Zhang, Huijian; Chen, Piaohua; Zhu, Yazhi; Zuazua Iriondo, Enrique (Structural Engineering and Mechanics, 2022)
We developed an accurate and simple vibration model to calculate the natural frequencies and their corresponding vibration modes for multi-span beam bridges with non-uniform cross-sections. A closed set of characteristic functions of a single-span beam was used to construct the vibration modes of the multi-span bridges, which were considered single-span beams with multiple constraints. To simplify the boundary conditions, the restraints were converted into spring constraints. Then the functional of the total energy has the same form as the penalty method. Compared to the conventional penalty method, the penalty coefficients in the proposed approach can be calculated directly, which can avoid the iteration process and convergence problem. The natural frequencies and corresponding vibration modes were obtained via the minimum total potential energy principle. By using the symmetry of the eigenfunctions or structure, the matrix size can be further reduced, which increases the computational efficiency of the proposed model. The accuracy and efficiency of the proposed approach were validated by the finite element method.
CRIS-Team Scopus Importer:2022-05-13
Keywords: Beam bridge; Bridge vibration; Multiple constraints; Natural frequency; Vibration mode

A Parabolic Approach to the Control of Opinion Spreading - Ruiz-Balet, Domènec; Zuazua Iriondo, Enrique (Applied Wave Mathematics II, 2019)
We analyse the problem of controlling to consensus a nonlinear system modelling opinion spreading. We derive explicit exponential estimates on the cost of approximately controlling these systems to consensus, as a function of the number of agents N and the control time horizon T. Our strategy makes use of known results on the controllability of spatially discretised semilinear parabolic equations. Both systems can be linked through time rescalin

A Potential Game Perspective in Federated Learning - Liu, Kang; Wang, Ziqi; Zuazua Iriondo, Enrique (2025)
Federated learning (FL) is an emerging paradigm for training machine learning models across distributed clients. Traditionally, in FL settings, a central server assigns training efforts (or strategies) to clients. However, from a market-oriented perspective, clients may independently choose their training efforts based on rational self-interest. To explore this, we propose a potential game framework where each client's payoff is determined by their individual efforts and the rewards provided by the server. The rewards are influenced by the collective efforts of all clients and can be modulated through a reward factor. Our study begins by establishing the existence of Nash equilibria (NEs), followed by an investigation of uniqueness in homogeneous settings. We demonstrate a significant improvement in clients' training efforts at a critical reward factor, identifying it as the optimal choice for the server. Furthermore, we prove the convergence of the best-response algorithm to compute NEs for our FL game. Finally, we apply the training efforts derived from specific NEs to a real-world FL scenario, validating the effectiveness of the identified optimal reward facto
https://cris.fau.de/converis/publicweb/Publication/331321390

Approximate and Weighted Data Reconstruction Attack in Federated Learning - Wang, Ziqi; Song, Yongcun; Zuazua Iriondo, Enrique (2024)
Federated Learning (FL) is a distributed learning paradigm that enables multiple clients to collaborate on building a machine learning model without sharing their private data. Although FL is considered privacy-preserved by design, recent data reconstruction attacks demonstrate that an attacker can recover clients' training data based on the parameters shared in FL. However, most existing methods fail to attack the most widely used horizontal Federated Averaging (FedAvg) scenario, where clients share model parameters after multiple local training steps. To tackle this issue, we propose an interpolation-based approximation method, which makes attacking FedAvg scenarios feasible by generating the intermediate model updates of the clients' local training processes. Then, we design a layer-wise weighted loss function to improve the data quality of reconstruction. We assign different weights to model updates in different layers concerning the neural network structure, with the weights tuned by Bayesian optimization. Finally, experimental results validate the superiority of our proposed approximate and weighted attack (AWA) method over the other state-of-the-art methods, as demonstrated by the substantial improvement in different evaluation metrics for image data reconstructions.<br
https://cris.fau.de/converis/publicweb/Publication/310887833

Approximating travelling waves by equilibria of non-local equations - Arrieta, Jose M.; Lopez-Fernandez, Maria; Zuazua, Enrique (Asymptotic Analysis, 2012)
We consider an evolution equation of parabolic type in R having a travelling wave solution. We study the effects on the dynamics of an appropriate change of variables which transforms the equation into a non-local evolution one having a travelling wave solution with zero speed of propagation with exactly the same profile as the original one. This procedure allows us to compute simultaneously the travelling wave profile and its propagation speed avoiding moving meshes, as we illustrate with several numerical examples. We analyze the relation of the new equation with the original one in the entire real line. We also analyze the behavior of the non-local problem in a bounded interval with appropriate boundary conditions. We show that it has a unique stationary solution which approaches the traveling wave as the interval gets larger and larger and that is asymptotically stable for large enough intervals.
CRIS-Team WoS Importer:2019-12-13

A quantitative analysis of Koopman operator methods for system identification and predictions - Zhang, Christophe; Zuazua Iriondo, Enrique (Comptes Rendus Mecanique, 2023)
We give convergence and cost estimates for a data-driven system identification method: given an unknown dynamical system, the aim is to recover its vector field and its flow from trajectory data. It is based on the so-called Koopman operator, which uses the well-known link between differential equations and linear transport equations. Data-driven methods recover specific finite-dimensional approximations of the Koopman operator, which can be understood as a transport operator. We focus on such approximations given by classical finite-elements spaces, which allow us to give estimates on the approximation of the Koopman operator as well as the solutions of the associated linear transport equation. These approximations are thus relevant objects to solve the system identification problem. We then analyze the convergence of a variant of the generator Extended Dynamic Mode Decom- position (gEDMD) algorithm, one of the main algorithms developed to compute approximations of the Koopman operator from data. We find however that, when combining this algorithm with clas- sical finite elements spaces, the results are not satisfactory numerically, as the convergence of the data-driven approximation is too slow for the method to benefit from the accuracy of finite elements spaces. In particular, for problems in dimension 1 it is less efficient than direct interpolation methods to recover the vector field. We provide some numerical examples to illustrate this last point.
Keywords: Koopman operator; system identification; finite elements spaces; data-driven approximation; Extended Dynamic Mode Decomposition

A Stochastic Approach to the Synchronization of Coupled Oscillators - Biccari, Umberto; Zuazua Iriondo, Enrique (Frontiers in Energy Research, 2020)
This paper deals with an optimal control problem associated with the Kuramoto model describing the dynamical behavior of a network of coupled oscillators. Our aim is to design a suitable control function allowing us to steer the system to a synchronized configuration in which all the oscillators are aligned on the same phase. This control is computed via the minimization of a given cost functional associated with the dynamics considered. For this minimization, we propose a novel approach based on the combination of a standard Gradient Descent (GD) methodology with the recently-developed Random Batch Method (RBM) for the efficient numerical approximation of collective dynamics. Our simulations show that the employment of RBM improves the performances of the GD algorithm, reducing the computational complexity of the minimization process and allowing for a more efficient control calculation.
CRIS-Team WoS Importer:2020-07-31

A stochastic approach to the synchronization of coupled oscillators - Biccari, Umberto; Zuazua Iriondo, Enrique (2020)
This paper deals with an optimal control problem associated to the Kuramoto model describing the dynamical behavior of a network of coupled oscillators. Our aim is to design a suitable control function allowing us to steer the system to a synchronized configuration in which all the oscillators are aligned on the same phase. This control is computed via the minimization of a given cost functional associated with the dynamics considered. For this minimization, we propose a novel approach based on the combination of a standard Gradient Descent (GD) methodology with the recently-developed Random Batch Method (RBM) for the efficient numerical approximation of collective dynamics. Our simulations show that the employment of RBM improves the performances of the GD algorithm, reducing the computational complexity of the minimization process and allowing for a more efficient control calculation.<br

Asymptotic behavior and control of a "guidance by repulsion" model - Ko, Dongnam; Zuazua Iriondo, Enrique (Mathematical Models & Methods in Applied Sciences, 2020)
We model and analyze a guiding problem, where the drivers try to steer the evaders' positions toward a target region while the evaders always try to escape from drivers. This problem is motivated by the guidance-by-repulsion model [R. Escobedo, A. Ibañez and E. Zuazua, Optimal strategies for driving a mobile agent in a "guidance by repulsion" model, Commun. onlinear Sci. umer. Simul. 39 (2016) 58-72] where the authors answer how to control the evader's position and what is the optimal maneuver of the driver. First, we analyze well posedness and behavior of the one-driver and one-evader model, assuming of the same friction coefficients. From the long-time behavior, the exact controllability is proved in a long enough time horizon. Then, we extend the model to the multi-driver and multi-evader case. We assumed three interaction rules in the context of collective behavior models: flocking between evaders, collision avoidance between drivers and repulsive forces between drivers and evaders. These interactions depend on the relative distances, and each agent is assumed to be undistinguishable and obtained an averaged effect from the other individuals. In this model, we develop numerical simulations to systematically explore the nature of controlled dynamics in various scenarios. The optimal strategies turn out to share a common pattern to the one-driver and one-evader case: the drivers rapidly occupy the position behind the target, and want to pursuit evaders in a straight line for most of the time. Inspired by this, we build a feedback strategy which stabilizes the direction of evaders.
CRIS-Team Scopus Importer:2020-03-31
Keywords: Asymptotic stability; collective behavior; guidance by repulsion; guiding problem; optimal control

Asymptotic expansions for anisotropic heat kernels - Ignat, Liviu I.; Zuazua, Enrique (Journal of Evolution Equations, 2013)
We obtain the asymptotic expansion of the solutions of some anisotropic heat equations when the initial data belong to polynomially weighted L (p) -spaces. We mainly address two model examples. In the first one, the diffusivity is of order two in some variables but higher in the other ones. In the second one, we consider the heat equation on the Heisenberg group.
CRIS-Team WoS Importer:2019-12-13

Asymptotic limits and stabilization for the 1D nonlinear Mindlin-Timoshenko system - Araruna, F. D.; Braz E Silva, Pablo; Zuazua, Enrique (Journal of Systems Science & Complexity, 2010)
This paper shows how the so called von Karman model can be obtained as a singular limit of a modified Mindlin-Timoshenko system when the modulus of elasticity in shear k tends to infinity, provided a regularizing term through a fourth order dispersive operator is added. Introducing damping mechanisms, the authors also show that the energy of solutions for this modified Mindlin-Timoshenko system decays exponentially, uniformly with respect to the parameter k. As k -> a, the authors obtain the damped von Karman model with associated energy exponentially decaying to zero as well.
CRIS-Team WoS Importer:2019-12-13

Asymptotics and stabilization for dynamic models of nonlinear beams - Araruna, Fagner D.; Braz E Silva, Pablo; Zuazua, Enrique (Proceedings of the Estonian Academy of Sciences, 2010)
We prove that the von Karman model for vibrating beams can be obtained as a singular limit of a modified Mindlin-Timoshenko system when the modulus of elasticity in shear k tends to infinity, provided a regularizing term through a fourth-order dispersive operator is added. We also show that the energy of solutions for this modified Mindlin-Timoshenko system decays exponentially, uniformly with respect to the parameter k, when suitable damping terms are added. As k -> infinity one deduces the uniform exponential decay of the energy of the von Karman model.
CRIS-Team WoS Importer:2019-12-13

A SYSTEMATIC METHOD FOR BUILDING SMOOTH CONTROLS FOR SMOOTH DATA - Ervedoza, Sylvain; Zuazua, Enrique (Discrete and Continuous Dynamical Systems-Series B, 2010)
We prove a regularity result for an abstract control problem z' = Az + Bv with initial datum z(0) = z(0) in which the goal is to determine a control v such that z(T) = 0. Under standard admissibility and observability assumptions on the adjoint system, when A generates a C(0) group, we develop a method to compute algorithmically a control function v that inherits the regularity of the initial datum to be controlled. In particular, the controlled equation is satisfied in a strong sense when the initial datum is smooth. In this way, the controlled trajectory is smooth as well. Our method applies mainly to time-reversible infinite-dimensional systems and, in particular, to the wave equation, but fails to be valid in the parabolic frame.
CRIS-Team WoS Importer:2019-12-13

A two-dimensional flea on the elephant phenomenon and its numerical visualization - Bianchini, Roberta; Gosse, Laurent; Zuazua, Enrique (Multiscale Modeling & Simulation, 2019)
Localization phenomena (sometimes called flea on the elephant) for the operator Lvarepsilon = varepsilon 2Δ u + p(x)u, p(x) being an asymmetric double well potential, are studied both analytically and numerically, mostly in two space dimensions within a perturbative framework. Starting from a classical harmonic potential, the effects of various perturbations are retrieved, especially in the case of two asymmetric potential wells. These findings are illustrated numerically by means of an original algorithm, which relies on a discrete approximation of the Steklov-Poincaré operator for Lvarepsilon, and for which error estimates are established. Such a two-dimensional discretization produces less mesh imprinting than more standard finite differences and correctly captures sharp layers.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Asymmetric double well potential; Bound states; Spectrum of Schrodinger equation

A Two-Stage Numerical Approach for the Sparse Initial Source Identification of a Diffusion-Advection Equation - Biccari, Umberto; Song, Yongcun; Yuan, Xiaomin; Zuazua Iriondo, Enrique (Inverse Problems, 2023)
We consider the problem of identifying a sparse initial source condition to achieve a given state distribution of a diffusion-advection partial differential equation after a given final time. The initial condition is assumed to be a finite combination of Dirac measures. The locations and intensities of this initial condition are required to be identified. This problem is known to be exponentially ill-posed because of the strong diffusive and smoothing effects. We propose a two-stage numerical approach to treat this problem. At the first stage, to obtain a sparse initial condition with the desire of achieving the given state subject to a certain tolerance, we propose an optimal control problem involving sparsity-promoting and ill-posedness-avoiding terms in the cost functional, and introduce a generalized primal-dual algorithm for this optimal control problem. At the second stage, the initial condition obtained from the optimal control problem is further enhanced by identifying its locations and intensities in its representation of the combination of Dirac measures. This two-stage numerical approach is shown to be easily implementable and its efficiency in short time horizons is promisingly validated by the results of numerical experiments. Some discussions on long time horizons are also included.<br
Keywords: initial source identification; inverse problem; optimal control; sparse control; diffusion-advection equations; non-smooth optimization; primal-dual algorithm

Averaged control and observation of parameter-depending wave equations - Lazar, Martin; Zuazua, Enrique (Comptes Rendus Mathematique, 2014)
We analyze the problem of averaged observability and control of wave equations.This topic is motivated by the control of parameter-dependent systems of wave equations. We look for controls ensuring the controllability of the averages of the states with respect to the parameter. This turns out to be equivalent to the problem of averaged observation in which one aims at recovering the energy of the initial data of the adjoint system by measurements done on its averages, under the assumption that the initial data of all the components of the adjoint system coincide.The problem under consideration is weaker than the classical notion of simultaneous observation and control.The method of proof uses propagation arguments based on H-measures or microlocal defect measures that reduce the problem to non-standard unique-continuation issues.Using transmutation techniques, we also derive some results on the averaged observation and control of parameter-dependent heat equations. © 2014 Académie des sciences.
CRIS-Team Scopus Importer:2021-06-15

Averaged control - Zuazua, Enrique (Automatica, 2014)
We analyze the problem of controlling parameter-dependent systems. We introduce the notion of averaged control according to which the quantity of interest is the average of the states with respect to the parameter. First we consider the problem of controllability for linear finite-dimensional systems and show that a necessary and sufficient condition for averaged controllability is an averaged rank condition, in the spirit of the classical rank condition for linear control systems, but involving averaged momenta of any order of the matrices generating the dynamics and representing the control action. We also describe some open problems and directions of possible research, in particular on the average controllability of evolution partial differential equations. In this context we analyze also the averaged version of a classical optimal control problem for a parameter dependent elliptic equation and derive the corresponding optimality system.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Averaged controllability; Averaged observability; Averaged optimal control; ODE and PDE; Parametrized

Averaged controllability for random evolution Partial Differential Equations - Lu, Qi; Zuazua, Enrique (Journal De Mathematiques Pures Et Appliquees, 2016)
We analyze the averaged controllability properties of random evolution Partial Differential Equations. We mainly consider heat and Schrödinger equations with random parameters, although the problem is also formulated in an abstract frame. We show that the averages of parabolic equations lead to parabolic-like dynamics that enjoy the null-controllability properties of solutions of heat equations in an arbitrarily short time and from arbitrary measurable sets of positive measure. In the case of Schrödinger equations we show that, depending on the probability density governing the random parameter, the average may behave either as a conservative or a parabolic-like evolution, leading to controllability properties, in average, of very different kind.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Averaged controllability; Averaged observability; Heat equation; Random evolution Partial Differential Equations; Schrödinger equation

Averaged controllability of parameter dependent conservative semigroups - Loheac, Jerome; Zuazua, Enrique (Journal of Differential Equations, 2017)
We consider the problem of averaged controllability for parameter depending (either in a discrete or continuous fashion) control systems, the aim being to find a control, independent of the unknown parameters, so that the average of the states is controlled. We do it in the context of conservative models, both in an abstract setting and also analysing the specific examples of the wave and Schrödinger equations. Our first result is of perturbative nature. Assuming the averaging probability measure to be a small parameter-dependent perturbation (in a sense that we make precise) of an atomic measure given by a Dirac mass corresponding to a specific realisation of the system, we show that the averaged controllability property is achieved whenever the system corresponding to the support of the Dirac is controllable. Similar tools can be employed to obtain averaged versions of the so-called Ingham inequalities. Particular attention is devoted to the 1d wave equation in which the time-periodicity of solutions can be exploited to obtain more precise results, provided the parameters involved satisfy Diophantine conditions ensuring the lack of resonances.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Averaged control; Ingham inequalities; Non-harmonic Fourier series; Parameter dependent systems; Perturbation arguments; Wave equation

Averaged dynamics and control for heat equations with random diffusion - Bárcena-Petisco, Jon Asier; Zuazua Iriondo, Enrique (Systems & Control Letters, 2021)
This paper deals with the averaged dynamics for heat equations in the degenerate case where the diffusivity coefficient, assumed to be constant, is allowed to take the null value. First we prove that the averaged dynamics is analytic. This allows to show that, most often, the averaged dynamics enjoys the property of unique continuation and is approximately controllable. We then determine if the averaged dynamics is actually null controllable or not depending on how the density of averaging behaves when the diffusivity vanishes. In the critical density threshold the dynamics of the average is similar to the [Formula presented]-fractional Laplacian, which is well-known to be critical in the context of the controllability of fractional diffusion processes. Null controllability then fails (resp. holds) when the density weights more (resp. less) in the null diffusivity regime than in this critical regime.
CRIS-Team Scopus Importer:2021-11-12
Keywords: Averaged controllability; Averaged observability; Observability; Random heat equation

Blowup for a time-oscillating nonlinear heat equation - Cazenave, Th; Escobedo, M.; Zuazua, Enrique (Acta Mathematica Universitatis Comenianae, 2013)
In this paper, we study a nonlinear heat equation with a periodic time oscillating term in factor of the nonlinearity. In particular, we give examples showing how the behavior of the solution can drastically change according to both the frequency of the oscillating factor and the size of the initial value.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Finite-time blowup; Semilinear heat equation; Time-oscillating nonlinearity

Boundary observation and control for fractional heat and wave equations - Biccari, Umberto; Warma, Mahamadi; Zuazua Iriondo, Enrique (2025)
We establish boundary observability and control for the fractional heat equation over arbitrary time horizons T>0, within the optimal range of fractional exponents s∈(1/2,1). Our approach introduces a novel synthesis of techniques from fractional partial differential equations and control theory, combining several key ingredients in an original and effective manner: 1. Boundary observability for low-frequency solutions of the fractional wave equation. We begin by analyzing the associated fractional wave equation. Using a fractional analogue of Pohozaev's identity, we establish a partial boundary observability result for the low-frequency solutions. The corresponding observability time horizon increases with the eigenmode frequency, reflecting the inherently slower propagation speed of the fractional waves. 2. Transmutation to the parabolic setting. Using transmutation techniques, we transfer the observability results from the wave setting to the parabolic one. This yields a frequency-dependent observability inequality for the fractional heat equation, which - via duality - enables control of its low-frequency components. 3. Frequency-wise iteration. Leveraging the dissipative nature of the fractional heat equation, we develop an iterative procedure to successively control the entire frequency spectrum of solutions. The condition s∈(1/2,1) is crucial in this analysis, as it guarantees sufficient decay of high-frequency components, enabling the convergence of the iteration. 4. Duality. By a duality argument, we derive boundary observability from the boundary controllability of the fractional heat equation. Remarkably, this type of boundary observability result is entirely new in the multi-dimensional setting and appears to be out of reach for existing method
https://cris.fau.de/converis/publicweb/Publication/339916906

Boundary sidewise observability of the wave equation - Dehman, Belhassen; Zuazua Iriondo, Enrique (2024)
The wave equation on a bounded domain of Rn with non homogeneous boundary Dirichlet data or sources supported on a subset of the boundary is considered. We analyze the problem of observing the source out of boundary measurements done away from its support. We first show that observability inequalities may not hold unless an infinite number of derivatives are lost, due to the existence of solutions that are arbitrarily concentrated near the source. We then establish observability inequalities in Sobolev norms, under a suitable microlo- cal geometric condition on the support of the source and the measurement set, for sources fulfilling pseudo-differential conditions that exclude these concentration phenomena. The proof relies on microlocal arguments and is essentially based on the use of microlocal defect measure
https://cris.fau.de/converis/publicweb/Publication/313264606

Boundary stabilization of numerical approximations of the 1-D variable coefficients wave equation: A numerical viscosity approach - Marica, Aurora; Zuazua, Enrique (Optimization with PDE Constraints, 2014)
In this paper, we consider the boundary stabilization problem associated to the 1- d wave equation with both variable density and diffusion coefficients and to its finite difference semi-discretizations. It is well-known that, for the finite difference semi-discretization of the constant coefficients wave equation on uniform meshes (Tébou and Zuazua, Adv. Comput. Math. 26:337–365, 2007) or on somenon-uniform meshes (Marica and Zuazua, BCAM, 2013, preprint), the discrete decay rate fails to be uniform with respect to the mesh-size parameter. We prove that, under suitable regularity assumptions on the coefficients and after adding an appropriate artificial viscosity to the numerical scheme, the decay rate is uniform as the mesh-size tends to zero. This extends previous results in Tébou and Zuazua (Adv. Comput.Math. 26:337–365, 2007) on the constant coefficient wave equation. The methodology of proof consists in applying the classical multiplier technique at the discrete level, with a multiplier adapted to the variable coefficients.
CRIS-Team Scopus Importer:2021-06-15
Keywords: 1-d wave equation; Artificial viscosity; Boundary stabilization; Variable density and diffusion coefficients

Classical system theory revisited for turnpike in standard state space systems and impulse controllable descriptor systems - Heiland, Jan; Zuazua Iriondo, Enrique (SIAM Journal on Control and Optimization, 2021)
The concept of turnpike connects the solution of long but finite time horizon optimal control problems with steady-state optimal controls. A key ingredient of the analysis of turnpike phenomena is the linear quadratic regulator problem and the convergence of the solution of the associated differential Riccati equation as the terminal time approaches infinity. This convergence has been investigated in linear systems theory in the 1980s. We extend classical system theoretic results for the investigation of turnpike properties of standard state space systems and descriptor systems. We present conditions for turnpike phenomena in the nondetectable case and for impulse controllable descriptor systems. For the latter, in line with the theory for standard linear systems, we establish existence and convergence of solutions to a generalized differential Riccati equation 2021 Society for Industrial and Applied Mathematics.
CRIS-Team Scopus Importer:2021-11-05
Keywords: Descriptor systems; Linear systems; Long time behavior; Optimal control; Riccati equations

Classical system theory revisited for Turnpike in standard state space systems and impulse controllable descriptor systems - Heiland, Jan; Zuazua Iriondo, Enrique (SIAM Journal on Control and Optimization, 2021)
The concept of turnpike connects the solution of long but finite time horizon optimal control problems with steady state optimal controls. A key ingredient of the analysis of turnpike phenomena is the linear quadratic regulator problem and the convergence of the solution of the associated differential Riccati equation as the terminal time approaches infinity. This convergence has been investigated in linear systems theory in the 1980s. We extend classical system theoretic results for the investigation of turnpike properties of standard state space systems and descriptor systems. We present conditions for turnpike phenomena in the non detectable case and for impulse controllable descriptor systems. For the latter, in line with the theory for standard linear systems,we establish existence and convergence of solutions to a generalized differential Riccati equatio
Keywords: linear systems; descriptor systems; optimal control; long time behavior; Riccati equations

Cluster-based classification with neural ODEs via control - Álvarez López, Antonio; Orive-Illera, Rafael; Zuazua Iriondo, Enrique (2025)
We address binary classification using neural ordinary differential equations from the perspective of simultaneous control of N data points. We consider a single-neuron architecture with parameters fixed as piecewise constant functions of time. In this setting, the model complexity can be quantified by the number of control switches. Previous work has shown that classification can be achieved using a point-by-point strategy that requires O(N) switches. We propose a new control method that classifies any arbitrary dataset by se- quentially steering clusters of d points, thereby reducing the complexity to O(N/d) switches. The optimality of this result, particularly in high dimensions, is supported by some numerical experiments. Our complexity bound is sufficient but often conservative because same-class points tend to appear in larger clusters, simpli- fying classification. This motivates studying the probability distribution of the number of switches required. We introduce a simple control method that imposes a collinearity constraint on the parameters, and analyze a worst-case scenario where both classes have the same size and all points are i.i.d. Our results highlight the benefits of high-dimensional spaces, showing that classification using constant controls becomes more probable as d increases
https://cris.fau.de/converis/publicweb/Publication/339724265

Clustering in Pure-Attention Hardmax Transformers and its Role in Sentiment Analysis - Alcalde Zafra, Albert; Fantuzzi, Giovanni; Zuazua Iriondo, Enrique (2024)
Transformers are extremely successful machine learning models whose mathematical properties remain poorly understood. Here, we rigorously characterize the behavior of transformers with hardmax self-attention and normalization sublayers as the number of layers tends to infinity. By viewing such transformers as discrete-time dynamical systems describing the evolution of points in a Euclidean space, and thanks to a geometric interpretation of the self-attention mechanism based on hyperplane separation, we show that the transformer inputs asymptotically converge to a clustered equilibrium determined by special points called leaders. We then leverage this theoretical understanding to solve sentiment analysis problems from language processing using a fully interpretable transformer model, which effectively captures `context' by clustering meaningless words around leader words carrying the most meaning. Finally, we outline remaining challenges to bridge the gap between the mathematical analysis of transformers and their real-life implementation. </block
https://cris.fau.de/converis/publicweb/Publication/324741614
Keywords: transformers; self-attention; clustering; AI interpretability; sentiment analysis

Complexity and regularity of maximal energy domains for the wave equation with fixed initial data - Privat, Yannick; Trelat, Emmanuel; Zuazua, Enrique (Discrete and Continuous Dynamical Systems, 2015)
We consider the homogeneous wave equation on a bounded open connected subset Ω of IRn. Some initial data being specified, we consider the problem of determining a measurable subset ω of Ω maximizing the L2-norm of the restriction of the corresponding solution to ω over a time interval [0, T], over all possible subsets of Ω having a certain prescribed measure. We prove that this problem always has at least one solution and that, if the initial data satisfy some analyticity assumptions, then the optimal set is unique and moreover has a finite number of connected components. In contrast, we construct smooth but not analytic initial conditions for which the optimal set is of Cantor type and in particular has an infinite number of connected components.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Calculus of variations.; Cantor set; Fourier series; Optimal domain; Wave equation

Computational performance of the MMOC in the inverse design of the Doswell frontogenesis equation - Francisco, Alexandre; Biccari, Umberto; Zuazua Iriondo, Enrique (2024)
Inverse design of transport equations can be addressed by using a gradient-adjoint methodology. In this methodology numerical schemes used for the adjoint resolution determine the direction of descent in its iterative algorithm, and consequently the CPU time consumed by the inverse design. As the CPU time constitutes a known bottleneck, it is important to employ light and quick schemes to the adjoint problem. In this regard, we proposed to use the Modified Method of Characteristics (MMOC). Despite not preserving identity conservation, the MMOC is computationally competitive. In this work we investigated the advantage of using the MMOC in comparison with the Lax-Friedrichs and Lax-Wendro? schemes for the inverse design problem. By testing the Doswell frontogenesis equation, we observed that the MMOC can provide more efficient and accurate computation under some simulation conditions.<br
https://cris.fau.de/converis/publicweb/Publication/317342604
Keywords: Characteristics-based method; inverse design; Doswell frontogenesis

Constrained control of bistable reaction-diffusion equations: Gene-flow and spatially heterogeneous models - Ruiz-Balet, Domènec; Mazari, Idriss; Zuazua Iriondo, Enrique (2019)
In this article, we study gene-flow models and the influence of spatial heterogeneity on the dynamics of bistable reaction-diffusion equations from the control point of view. We establish controllability results under geometric assumptions on the domain where the system evolves and regularity assumptions on the spatial heterogeneity. The non-linearity is assumed to be of bistable type and to have three spatially homogeneous equilibria. We investigate whether or not it is possible, starting from any initial datum, to drive the population to one of these equilibria through a boundary control u, under the natural constraints 0 u 1. In the case of the gene-flow model, the situation is similar to [35, 33] and the results only depend on the geometry of the domain. In the case of a spatially heterogeneous environment, we distinguish between slowly varying environments and of rapidly varying ones. We develop a new method to prove that controllability to 0, θ or 1 still holds in the slowly varying case, and give examples of rapidly varying environments where controllability to 0 or 1 no longer holds. This lack of controllability is established by studying the existence of non-trivial solutions which act as barriers for controlling the dynamics, which are of independent interest. Our article is completed by several numerical exepriments that confirm our analysis.<br
Keywords: Control; reaction-diffusion equations; bistable equations; spatial heterogeneity; geneflow models; staircase method

Constrained control of gene-flow models - Mazari, Idriss; Ruiz-Balet, Domènec; Zuazua Iriondo, Enrique (Annales de l'Institut Henri Poincaré - Analyse Non Linéaire, 2023)
In ecology and population dynamics, gene flow refers to the transfer of a trait (e.g. genetic material) from one population to another. This phenomenon is of great relevance in studying the spread of diseases or the evolution of social features, such as languages. From the mathematical point of view, gene flow is modeled using bistable reaction-diffusion equations. The unknown is the proportion p of the population that possesses a certain trait, within an overall population N. In such models, gene flow is taken into account by assuming that the population density N depends either on p (if the trait corresponds to fitter individuals) or on the location x (if some zones in the domain can carry more individuals). Recent applications stemming from mosquito-borne-disease control problems or from the study of bilingualism have called for the investigation of the controllability properties of these models. At the mathematical level, this corresponds to boundary control problems and, since we are working with proportions, the control u has to satisfy the constraint 0 6 u 6 1. In this article, we provide a thorough analysis of the influence of the gene-flow effect on boundary controllability properties. We prove that, when the population density N only depends on the trait proportion p, the geometry of the domain is the only criterion that has to be considered. We then tackle the case of population densities N varying in x. We first prove that, when N varies slowly in x and when the domain is narrow enough, controllability always holds. This result is proved using a robust domain perturbation method. We then consider the case of sharp fluctuations in N: we first give examples that prove that controllability may fail. Conversely, we give examples of heterogeneities N such that controllability will always be guaranteed: in other words the controllability properties of the equation are very strongly influenced by the variations of N. All negative controllability results are proved by showing the existence of nontrivial stationary states, which act as barriers. The existence of such solutions and the methods of proof are of independent interest. Our article is completed by several numerical experiments that confirm our analysis.
CRIS-Team Scopus Importer:2023-08-25
Keywords: bistable equations; Control; gene-flow models; noncontrollability; reaction-diffusion equations; spatial heterogeneity; staircase method

Constrained control of gene-flow models - Mazari, Idriss; Ruiz-Balet, Domènec; Zuazua Iriondo, Enrique (2024)
In ecology and population dynamics, gene-flow refers to the transfer of a trait (e.g. genetic material) from one population to another. This phenomenon is of great relevance in studying the spread of diseases or the evolution of social features, such as languages. From the mathematical point of view, gene-flow is modelled using bistable reaction-diffusion equations. The unknown is the proportion p of the population that possesses a certain trait, within an overall population N. In such models, gene-flow is taken into account by assuming that the population density N depends either on p (if the trait corresponds to fitter individuals) or on the location x (if some zones in the domain can carry more individuals). Recent applications stemming from mosquito-borne disease control problems or from the study of bilingualism have called for the investigation of the controllability properties of these models. At the mathematical level, this corresponds to boundary control problems and, since we are working with proportions, the control u has to satisfy the constraints 0≤u≤1 0 ≤ u ≤ 10≤u≤1. In this article, we provide a thorough analysis of the influence of the gene-flow effect on boundary controllability properties. We prove that, when the population density N only depends on the trait proportion p, the geometry of the domain is the only criterion that has to be considered. We then tackle the case of population densities N varying in x. We first prove that, when N varies slowly in x and when the domain is narrow enough, controllability always holds. This result is proved using a robust domain perturbation method. We then consider the case of sharp fluctuations in N: we first give examples that prove that controllability may fail. Conversely, we give examples of heterogeneities N such that controllability will always be guaranteed: in other words the controllability properties of the equation are very strongly in uenced by the variations of N. All negative controllability results are proved by showing the existence of non-trivial stationary states, which act as barriers. The existence of such solutions and the methods of proof are of independent interest. Our article is completed by several numerical experiments that confirm our analysis.<br
https://cris.fau.de/converis/publicweb/Publication/267739598
Keywords: Control; reaction-diffusion equations; bistable equations; spatial heterogeneity; gene-flow models; staircase method; non-controllability

Continuous adjoint approach for the spalart-allmaras model in aerodynamic optimization - Bueno-Orovio, Alfonso; Castro, Carlos; Palacios, Francisco; Zuazua, Enrique (Aiaa Journal, 2012)
In this paper, the continuous adjoint method to compute shape sensitivities in aerodynamic design with turbulence modeling is described and developed. The focus is on compressible flows described by the Reynolds-averaged Navier-Stokes equations and the classical Spalart-Allmaras model for turbulence. Turbulence modeling usually requires, in particular, computation of the distance to the surface. Here, this distance is incorporated to the system as a new variable, solving the Eikonal equation. The accuracy of the sensitivity derivatives obtained with the complete turbulent approach is assessed by comparison with finite difference computations and the classical continuous adjoint with frozen viscosity, showing substantial improvements in the convergence properties of the method and in the quality of the obtained gradients. The validity of the overall methodology is illustrated with several design examples, including the optimization of three-dimensional geometries in combination with advanced freeform techniques for mesh deformation. Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
CRIS-Team Scopus Importer:2021-06-15

CONTROL AND ANALYSIS OF PARTIAL DIFFERENTIAL EQUATIONS - Cindea, Nicolae; Leugering, Günter; Loheac, Jerome; Micu, Sorin; Iacob, Gheorghe Mihoc-Caius; Morris, Kirsten; Takahashi, Takeo; Zuazua Iriondo, Enrique (2022)
CRIS-Team WoS Importer:2022-11-25

CONTROL AND ANALYSIS OF PARTIAL DIFFERENTIAL EQUATIONS - Cîndea, Nicolae; Leugering, Günter; Lohéac, Jérôme; Micu, Sorin; Morris, Kirsten; Takahashi, Takéo; Zuazua Iriondo, Enrique (Mathematical Reports, 2022)
CRIS-Team Scopus Importer:2024-12-16

Control and Machine Learning - Zuazua Iriondo, Enrique (SIAM News, 2022)
In this article, we shall present some recent results on the interplay between control and Machine Learning, and more precisely, Supervised Learning and Universal Approximation. We adopt the perspective of the simultaneous or ensemble control of systems of Residual Neural Networks (ResNets). Roughly, each item to be classified corresponds to a different initial datum for the Cauchy problem of the ResNets, leading to an ensemble of solutions to be driven to the corresponding targets, associated to the labels, by means of the same control. We present a genuinely nonlinear and constructive method, allowing to show that such an ambitious goal can be achieved, estimating the complexity of the control strategies. This property is rarely fulfilled by the classical dynamical systems in Mechanics and the very nonlinear nature of the activation function governing the ResNet dynamics plays a determinant role. It allows deforming half of the phase space while the other half remains invariant, a property that classical models in mechanics do not fulfill. The turnpike property is also analyzed in this context, showing that a suitable choice of the cost functional used to train the ResNet leads to more stable and robust dynamics.<br
Keywords: control; machine learning; ML

Control and Machine Learning - Zuazua Iriondo, Enrique (2024)
In this talk, we discuss recent results that explore the relationship between control theory and machine learning, specifically applied to supervised learning. First, we study the classification and approximation properties of residual neural networks. Interpreting these problems as simultaneous or ensemble control ones, we build genuinely nonlinear and constructive algorithms, estimating the complexity of controls. Then, we analyze the multilayer perceptron architecture, characterizing the necessary depth and minimum width required to achieve simultaneous controllability. In the domain of large language models, residual networks are combined in an alternating manner with self-attention layers, whose role is to capture the “context”. We view these layers as a dynamical system acting on a collection of points and characterize their asymptotic dynamics and convergence towards special points called leaders. We use our theoretical results to design an interpretable model to solve the task of sentiment analysis of movie reviews. Finally, we investigate federated learning, which enables multiple clients to collaboratively train models without sharing private data, thereby addressing data collection and privacy challenges. Within this framework, we address issues related to training efficiency, incentive mechanisms, and privacy concerns.
https://cris.fau.de/converis/publicweb/Publication/325045212

Control and Numerical approximation of Fractional Diffusion Equations - Biccari, Umberto; Warma, Mahamadi; Zuazua Iriondo, Enrique (Handbook of Numerical Analysis, 2022)
Abstract. The aim of this work is to give a broad panorama of the control properties of fractional diffusive models from a numerical analysis and simulation perspective. We do this by surveying several research results we obtained in the last years, focusing in particular on the numerical computation of controls, though not forgetting to recall other relevant contributions which can be currently found in the literature of this prolific field. Our reference model will be a non-local diffusive dynamics driven by the fractional Laplacian on a bounded domain Ω. The starting point of our analysis will be a Finite Element approximation for the associated elliptic model in one and two space-dimensions, for which we also present error estimates and convergence rates in the L^2 and energy norm. Secondly, we will address two specific control scenarios: firstly, we consider the standard interior control problem, in which the control is acting from a small subset ω⊂Ω. Secondly, we move our attention to the exterior control problem, in which the control region O⊂Ωc is located outside Ω. This exterior control notion extends boundary control to the fractional framework, in which the non-local nature of the models does not allow for controls supported on ∂Ω. We will conclude by discussing the interesting problem of simultaneous control, in which we consider families of parameter-dependent fractional heat equations and we aim at designing a unique control function capable of steering all the different realizations of the model to the same target configuration. In this framework, we will see how the employment of stochastic optimization techniques may help in alleviating the computational burden for the approximation of simultaneous controls. Our discussion is complemented by several open problems related with fractional models which are currently unsolved and may be of interest for future investigatio
Keywords: Fractional Laplacian; Fractional diffusion equation; Finite Element Method; Interior Control; Exterior Control; Simultaneous Control; Numerical approximation

Control and stabilization of waves on 1-d networks - Zuazua, Enrique (Modelling and Optimisation of Flows on Networks, 2013)
We present some recent results on control and stabilization of waves on 1-d networks.The fine time-evolution of solutions of wave equations on networks and, consequently, their control theoretical properties, depend in a subtle manner on the topology of the network under consideration and also on the number theoretical properties of the lengths of the strings entering in it. Therefore, the overall picture is quite complex.In this paper we summarize some of the existing results on the problem of controllability that, by classical duality arguments in control theory, can be reduced to that of observability of the adjoint uncontrolled system. The problem of observability refers to that of recovering the total energy of solutions by means of measurements made on some internal or external nodes of the network. They lead, by duality, to controllability results guaranteeing that L 2-controls located on those nodes may drive sufficiently smooth solutions to equilibrium at a final time. Most of our results in this context, obtained in collaboration with R. Dáger, refer to the problem of controlling the network from one single external node. It is, to some extent, the most complex situation since, obviously, increasing the number of controllers enhances the controllability properties of the system. Our methods of proof combine sidewise energy estimates (that in the particular case under consideration can be derived by simply applying the classical d'Alembert's formula), Fourier series representations, non-harmonic Fourier analysis, and number theoretical tools.These control results belong to the class of the so-called open-loop control systems.We then discuss the problem of closed-loop control or stabilization by feedback. We present a recent result, obtained in collaboration with J. Valein, showing that the observability results previously derived, regardless of the method of proof employed, can also be recast a posteriori in the context of stabilization, so to derive explicit decay rates (as) for the energy of smooth solutions. The decay rate depends in a very sensitive manner on the topology of the network and the number theoretical properties of the lengths of the strings entering in it.In the end of the article we also present some challenging open problems. © 2013 Springer-Verlag Berlin Heidelberg.
CRIS-Team Scopus Importer:2021-06-15

Controllability and positivity constraints in population dynamics with age structuring and diffusion - Maity, Debayan; Tucsnak, Marius; Zuazua, Enrique (Journal De Mathematiques Pures Et Appliquees, 2019)
In this article, we study the null controllability of a linear system coming from a population dynamics model with age structuring and spatial diffusion (of Lotka–McKendrick type). The control is localized in the space variable as well as with respect to the age. The first novelty we bring in is that the age interval in which the control needs to be active can be arbitrarily small and does not need to contain a neighbourhood of 0. The second one is that we prove that the whole population can be steered into zero in a uniform time, without, as in the existing literature, excluding some interval of low ages. Moreover, we improve the existing estimates of the controllability time and we show that our estimates are sharp, at least when the control is active for very low ages. Finally, we show that the system can be steered between two positive steady states by controls preserving the positivity of the state trajectory. The method of proof, combining final-state observability estimates with the use of characteristics and with L∞ estimates of the associated semigroup, avoids the explicit use of parabolic Carleman estimates.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Controllability with positivity constraints; Null controllability; Population dynamics

Controllability of a class of infinite dimensional systems with age structure - Maity, Debayan; Tucsnak, Marius; Zuazua Iriondo, Enrique (Control and Cybernetics, 2019)
Given a linear dynamical system, we investigate the linear infinite dimensional system obtained by grafting an age structure. Such systems appear essentially in population dynamics with age structure when phenomena like spatial diffusion or transport are also taken into consideration. We first show that the new system preservessome of the wellposedness properties of the initial one. Our main result asserts that if the initial system is null controllable in a time small enough then the structured system is also null controllable in a time depending on the various involved parameters.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Admissible control operator; Age structure; Infinite dimensional linear system; Null controllability; Population dynamics

Controllability of one-dimensional viscous free boundary flows - Geshkovski, Borjan; Zuazua Iriondo, Enrique (2019)
In this work, we address the local controllability of a one-dimensional free boundary problem for a fluid governed by the viscous Burgers equation. The free boundary manifests itself as one moving end of the interval, and its evolution is given by the value of the fluid velocity at this endpoint. We prove that, by means of a control actuating along the fixed boundary, we may steer the fluid to constant velocity in addition to prescribing the free boundary’s position, provided the initial velocities and interface positions are close enough. <br
Keywords: Controllability; free boundary problem; viscous Burgers equation

Controllability of one-dimensional viscous free boundary flows - Geshkovski, Borjan; Zuazua Iriondo, Enrique (SIAM Journal on Control and Optimization, 2021)
In this work, we address the local controllability of a one-dimensional free boundary problem for a fluid governed by the viscous Burgers equation. The free boundary manifests itself as one moving end of the interval, and its evolution is given by the value of the fluid velocity at this endpoint. We prove that, by means of a control actuating along the fixed boundary, we may steer the fluid to constant velocity in addition to prescribing the free boundary's position, provided the initial velocities and interface positions are close enough.
CRIS-Team Scopus Importer:2021-06-04
Keywords: Controllability; Free boundary problem; Viscous Burgers equation

Controllability of shadow reaction-diffusion systems - Hernández-Santamaría, Víctor; Zuazua Iriondo, Enrique (Journal of Differential Equations, 2019)
We study the null controllability of linear shadow models for reaction-diffusion systems arising as singular limits when the diffusivity of some of the components is very high. This leads to a coupled system where one component solves a parabolic partial differential equation (PDE) and the other one an ordinary differential equation (ODE). We analyze these shadow systems from a controllability perspective and prove two types of results. First, by employing Carleman inequalities and ODE arguments, we prove that the null controllability of the shadow model holds. This result, together with the effectiveness of the controls to control the original dynamics, is illustrated by numerical simulations. We also obtain a uniform Carleman estimate for the reaction-diffusion equations which allows to obtain the null control for the shadow system as a limit when the diffusivity tends to infinity in one of the equations.
CRIS-Team Scopus Importer:2019-12-03
Keywords: Carleman estimates; Hybrid systems; Neumann boundary conditions; Null controllability; Reaction-diffusion systems; Shadow model

Controllability of the Kirchhoff system for beams as a limit of the Mindlin-Timoshenko system - Araruna, F. D.; Zuazua, Enrique (SIAM Journal on Control and Optimization, 2008)
We consider the dynamical one-dimensional Mindlin-Timoshenko system for beams. We analyze how its controllability properties depend on the modulus of elasticity in shear k. In particular we prove that the exact boundary controllability property of the Kirchhoff system may be obtained as a singular limit, as k -> infinity, of the partial controllability of projections over a sharp subspace of solutions generated by the eigenfunctions that converge, as k -> infinity, towards the spectrum of the Kirchhoff system.
CRIS-Team WoS Importer:2019-12-13

Controllability of the one-dimensional fractional heat equation under positivity constraints - Biccari, Umberto; Warma, Mahamadi; Zuazua Iriondo, Enrique (Communications on Pure and Applied Analysis, 2020)
In this paper, we analyze the controllability properties under positivity constraints on the control or the state of a one-dimensional heat equation involving the fractional Laplacian (-d(x)(2))(s) (0 < s < 1) on the interval (-1, 1). We prove the existence of a minimal (strictly positive) time T-min such that the fractional heat dynamics can be controlled from any initial datum in L-2 (-1, 1) to a positive trajectory through the action of a positive control, when s > 1/2. Moreover, we show that in this minimal time constrained controllability is achieved by means of a control that belongs to a certain space of Radon measures. We also give some numerical simulations that confirm our theoretical results.
CRIS-Team WoS Importer:2020-01-31

Controllability Under Positivity Constraints of Multi-d Wave Equations - Pighin, Dario; Zuazua, Enrique (Trends in Control Theory and Partial Differential Equations, 2019)
We consider both the internal and boundary controllability problems for wave equations under non-negativity constraints on the controls. First, we prove the steady state controllability property with nonnegative controls for a general class of wave equations with time-independent coefficients. According to it, the system can be driven from a steady state generated by a strictly positive control to another, by means of nonnegative controls, and provided the time of control is long enough. Secondly, under the added assumption of conservation and coercivity of the energy, controllability is proved between states lying on two distinct trajectories. Our methods are described and developed in an abstract setting, to be applicable to a wide variety of control systems.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Bounded trajectories; Constraints; Controllability; Energy conservation; Quasi-static iteration; Wave equations

Controllability under positivity constraints of semilinear heat equations - Pighin, Dario; Zuazua, Enrique (Mathematical Control and Related Fields, 2018)
In many practical applications of control theory some constraints on the state and/or on the control need to be imposed. In this paper, we prove controllability results for semilinear parabolic equations under positivity constraints on the control, when the time horizon is long enough. As we shall see, in fact, the minimal controllability time turns out to be strictly positive. More precisely, we prove a global steady state constrained controllability result for a semilinear parabolic equation with C1 nonlinearity, without sign or globally Lipschitz assumptions on the nonlinear term. Then, under suitable dissipativity assumptions on the system, we extend the result to any initial datum and any target trajectory. We conclude with some numerical simulations that confirm the theoretical results that provide further information of the sparse structure of constrained controls in minimal time.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Controllability under constraints; Dissipativity; Iterative method; Semilinear heat equations; Waiting time

Control of 2D scalar conservation laws in the presence of shocks - Lecaros, Rodrigo; Zuazua, Enrique (Mathematics of Computation, 2016)
We analyze a model optimal control problem for a 2D scalar conservation law-the so-called inverse design problem-with the goal being to identify the initial datum leading to a given final time configuration. The presence of shocks is an impediment for classical methods, based on linearization, to be directly applied. We develop an alternating descent method that exploits the generalized linearization that takes into account both the sensitivity of the shock location and of the smooth components of solutions. A numerical implementation is proposed using splitting and finite differences. The descent method we propose is of alternating nature and combines variations taking account of the shock location and those that take care of the smooth components of the solution. The efficiency of the method is illustrated by numerical experiments.
CRIS-Team Scopus Importer:2021-06-15

Control of a Lotka-Volterra System with Weak Competition - Sônego, Maicon; Zuazua Iriondo, Enrique (2024)
The Lotka-Volterra model reflects real ecological interactions where species com- pete for limited resources, potentially leading to coexistence, dominance of one species, or extinction of another. Comprehending the mechanisms governing these systems can yield critical insights for developing strategies in ecological management and biodiversity conservation. In this work, we investigate the controllability of a Lotka-Volterra system model- ing weak competition between two species. Through constrained controls acting on the boundary of the domain, we establish conditions under which the sys- tem can be steered towards various target states. More precisely, we show that the system is controllable in finite time towards a state of coexistence whenever it exists, and asymptotically controllable towards single-species states or total extinction, depending on domain size, diffusion, and competition rates. Addition- ally, we determine scenarios where controllability is not possible and, in these cases, we construct barrier solutions that prevent the system from reaching spe- cific targets. Our results offer critical insights into how competition, diffusion, and spatial domain influence species dynamics under constrained controls. Sev- eral numerical experiments complement our analysis, confirming the theoretical findings.
https://cris.fau.de/converis/publicweb/Publication/329584013
Keywords: Lotka-Volterra; Boundary Control; Weak Competition; Barrier Solutions

Control of certain parabolic models from biology and social sciences - Ruiz-Balet, Domènec; Zuazua Iriondo, Enrique (Mathematical Control and Related Fields, 2022)
These lecture notes address the controllability under relevant state constraints of reaction-diffusion equations. Typically the quantities modeled by reaction-diffusion equations in socio-biological contexts (e.g. population, concentrations of chemicals, temperature, proportions etc.) are positive by nature. The uncontrolled models intrinsically preserve this nature thanks to the maximum principle. For this reason, any control strategy for such systems has to preserve these state constraints. We restrict our study in the case of scalar equations with monostable and bistable nonlinearities. The presence of constraints produces new phenomena such as a possible lack of controllability, or existence of a minimal controllability time. Furthermore, we explain general ways for proving controllability under state constraints. Among different strategies, we discuss how to use traveling waves and connected paths of steady states to ensure controllability. We devote particular attention to the construction of such connected paths of steady-states. Further recent extensions are presented, and open problems are settled. All the discussions are complemented with numerical simulations to provide intuition to the reader.
Keywords: Reaction-diffusion; Control; Steady-states; Constraints; Mathematical Biology

Control of Hyperbolic and Parabolic Equations on Networks and Singular limits - Bárcena-Petisco, Jon Asier; Cavalcante, M; Coclite, Giuseppe Maria; De Nitti, Nicola; Zuazua Iriondo, Enrique (Mathematical Control and Related Fields, 2025)
We study the controllability properties of the transport equation and of parabolic equations posed on a tree. Using a control localized on the exterior nodes, we prove that the hyperbolic and the parabolic systems are null-controllable. The hyperbolic proof relies on the method of characteristics, the parabolic one on duality arguments and Carleman inequalities. We also show that the parabolic system may not be controllable if we do not act on all exterior vertices because of symmetries. Moreover, we estimate the cost of the null-controllability of transport-diffusion equations with diffusivity ε > 0 and study its asymptotic behavior when ε → 0 +. We prove that the cost of the controllability decays for a time sufficiently large and explodes for short times. This is done by duality arguments allowing to reduce the problem to obtain observability estimates which depend on the viscosity parameter. These are derived by using Agmon and Carleman inequalities.<br
CRIS-Team Scopus Importer:2025-01-31
Keywords: Controllability; cost of controllability; advection-diffusion equations; singular limits; vanishing viscosity; networks

Control of neural transport for normalising flows - Ruiz-Balet, Domènec; Zuazua Iriondo, Enrique (Journal De Mathematiques Pures Et Appliquees, 2024)
Inspired by normalizing flows, we analyze the bilinear control of neural transport equations by means of time-dependent velocity fields restricted to fulfill, at any time instance, a simple neural network ansatz. The L^1 approximate controllability property is proved, showing that any probability density can be driven arbitrarily close to any other one in any time horizon. The control vector fields are built explicitly and inductively and this provides quantitative estimates on their complexity and amplitude. This also leads to statistical error bounds when only random samples of the target probability density are available.<br

Control of reaction-diffusion models in biology and social sciences - Ruiz-Balet, Domènec; Zuazua Iriondo, Enrique (2024)
These lecture notes address the controllability under state constraints of reaction-diffusion equations arising in socio-biological contexts. We restrict our study to scalar equations with monostable and bistable nonlinearities. The uncontrolled models describing, for instance, population dynamics, concentrations of chemicals, temperatures, etc., intrinsically preserve pointwise bounds of the states that represent a proportion, volume-fraction, or density. This is guaranteed, in the absence of control, by the maximum or comparison principle. We focus on the classical controllability problem, in which one aims to drive the system to a final target, for instance, a steady-state. In this context the state is required to preserve, in the presence of controls, the pointwise bounds of the uncontrolled dynamics. The presence of constraints introduces significant added complexity for the control process. They may force the needed control-time to be large enough or even make some natural targets to be unreachable, due to the presence of barriers that the controlled trajectories might not be able to overcome. We develop and present a general strategy to analyze these problems. We show how the combination of the various intrinsic qualitative properties of the systems’ dynamics and, in particular, the use of traveling waves and steady-states’ paths, can be employed to build controls driving the system to the desired target. We also show how, depending on the value of the Allee parameter and on the size of the domain in which the process evolves, some natural targets might become unreachable. This is consistent with empirical observations in the context of endangered minoritized languages and species at risk of extinction. Further recent extensions are presented, and open problems are settled. All the discussions are complemented with numerical simulations to illustrate the main methods and result
https://cris.fau.de/converis/publicweb/Publication/276859119
Keywords: Reaction-diffusion; control; paths of steady-states; constraints; phase plane; traveling waves; comparison principle; Mathematical Biology

Control under constraints for multi-dimensional reaction-diffusion monostable and bistable equations - Ruiz-Balet, Domènec; Zuazua Iriondo, Enrique (2019)
Dynamic phenomena in social and biological sciences can often be modeled by employing reactiondiffusion equations. Frequently in applications, their control plays an important role when avoiding population extinction or propagation of infectious diseases, enhancing multicultural features, etc. When addressing these issues from a mathematical viewpoint one of the main challenges is that, because of the intrinsic nature of the models under consideration, the solution, typically a proportion or a density function, needs to preserve given lower and upper bounds (taking values in [0, 1])). Controlling the system to the desired final configuration then becomes complex, and sometimes even impossible. In the present work, we analyze the controllability to constant steady-states of spatially homogeneous semilinear heat equations, with constraints in the state, and using boundary controls, which is indeed a natural way of acting on the system in the present context. The nonlinearities considered are among the most frequent: monostable and bistable ones. We prove that controlling the system to a constant steady-state may become impossible when the diffusivity is too small (or when the domain is large), due to the existence of barrier functions. When such an obstruction does not arise, we build sophisticated control strategies combining the dissipativity of the system, the existence of traveling waves and some connectivity of the set of steady-states. This connectivity allows building paths that the controlled trajectories can follow, in a long time, with small oscillations, preserving the natural constraints of the system. This kind of strategy was successfully implemented in one-space dimension, where phase plane analysis techniques allowed to decode the nature of the set of steady-states. These techniques fail in the present multi-dimensional setting. We employ a fictitious domain technique, extending the system to a larger ball, and building paths of radially symmetric solution that can then be restricted to the original domain. The results are illustrated by numerical simulations of these models that find several applications, such as the extinction of minority languages or the survival of rare species in sufficiently large reserved areas.<br
Keywords: Controllability; Reaction-diffusion; Constraints; Mathematical Biology

Control under constraints for multi-dimensional reaction-diffusion monostable and bistable equations - Ruiz-Balet, Domènec; Zuazua Iriondo, Enrique (Journal De Mathematiques Pures Et Appliquees, 2020)
Dynamic phenomena in social and biological sciences can often be modeled by reaction-diffusion equations. When addressing the control from a mathematical viewpoint, one of the main challenges is that, because of the intrinsic nature of the models under consideration, the solution, typically a proportion or a density function, needs to preserve given lower and upper bounds (taking values in [0,1])). Controlling the system to the desired final configuration then becomes complex, and sometimes even impossible. In the present work, we analyze the controllability to constant steady states of spatially homogeneous monostable and bistable semilinear heat equations, with constraints in the state, and using boundary controls. We prove that controlling the system to a constant steady state may become impossible when the diffusivity is too small due to the existence of barrier functions. We build sophisticated control strategies combining the dissipativity of the system, the existence of traveling waves, and some connectivity of the set of steady states to ensure controllability whenever it is possible. This connectivity allows building paths that the controlled trajectories can follow, in a long time, with small oscillations, preserving the natural constraints of the system. This kind of strategy was successfully implemented in one-space dimension, where phase plane analysis techniques allowed to decode the nature of the set of steady states. These techniques fail in the present multi-dimensional setting. We employ a fictitious domain technique, extending the system to a larger ball, and building paths of radially symmetric solution that can then be restricted to the original domain.
CRIS-Team Scopus Importer:2020-09-04
Keywords: Constraints; Controllability; Mathematical biology; Reaction-diffusion

Convergence of a two-grid algorithm for the control of the wave equation - Ignat, Liviu I.; Zuazua, Enrique (Journal of the European Mathematical Society, 2009)
We analyze the problem of boundary observability of the finite-difference space semidiscretizations of the 2-d wave equation in the square. We prove the uniform (with respect to the mesh-size) boundary observability for the solutions obtained by the two-grid preconditioner introduced by Glowinski [9]. Our method uses previously known uniform observability inequalities for low frequency solutions and a dyadic spectral time decomposition. As a consequence we prove the convergence of the two-grid algorithm for computing the boundary controls for the wave equation. The method can be applied in any space dimension, for more general domains and other discretization schemes.
CRIS-Team WoS Importer:2019-12-13

Convergence rates for dispersive approximation schemes to nonlinear Schrodinger equations - Ignat, Liviu I.; Zuazua, Enrique (Journal De Mathematiques Pures Et Appliquees, 2012)
This article is devoted to the analysis of the convergence rates of several numerical approximation schemes for linear and nonlinear Schrodinger equations on the real line. Recently, the authors have introduced viscous and two-grid numerical approximation schemes that mimic at the discrete level the so-called Strichartz dispersive estimates of the continuous Schrodinger equation. This allows to guarantee the convergence of numerical approximations for initial data in L-2(R), a fact that cannot be proved in the nonlinear setting for standard conservative schemes unless more regularity of the initial data is assumed. In the present article we obtain explicit convergence rates and prove that dispersive schemes fulfilling the Strichartz estimates are better behaved for H-s (R) data if 0 < s < 1/2. Indeed, while dispersive schemes ensure a polynomial convergence rate, non-dispersive ones only yield logarithmic ones. (C) 2012 Elsevier Masson SAS. All rights reserved.
CRIS-Team WoS Importer:2019-12-13

Decay rates for 1-d heat-wave planar networks - Han, Zhong-Jie; Zuazua, Enrique (Networks and Heterogeneous Media, 2016)
The large time decay rates of a transmission problem coupling heat and wave equations on a planar network is discussed. When all edges evolve according to the heat equation, the uniform exponential decay holds. By the contrary, we show the lack of uniform stability, based on a Geometric Optics high frequency asymptotic expansion, whenever the network involves at least one wave equation. The (slow) decay rate of this system is further discussed for star-shaped networks. When only one wave equation is present in the network, by the frequency domain approach together with multipliers, we derive a sharp polynomial decay rate. When the network involves more than one wave equation, a weakened observability estimate is obtained, based on which, polynomial and logarithmic decay rates are deduced for smooth initial conditions under certain irrationality conditions on the lengths of the strings entering in the network. These decay rates are intrinsically determined by the wave equations entering in the system and are independent on the heat equations.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Asymptotic behavior; irrationality conditions; Geometric Optics; Hyperbolic-parabolic coupled system; Logarithmic decay; Observability; Polynomial decay; Star-shaped network

Decay rates for elastic-thermoelastic star-shaped networks - Han, Zhong-Jie; Zuazua, Enrique (Networks and Heterogeneous Media, 2017)
This work discusses the asymptotic behaviour of a transmission problem on star-shaped networks of interconnected elastic and thermoelastic rods. Elastic rods are undamped, of conservative nature, while the thermoelastic ones are damped by thermal effects. We analyse the overall decay rate depending of the number of purely elastic components entering on the system and the irrationality properties of its lengths. First, a sufficient and necessary condition for the strong stability of the thermoelastic-elastic network is given. Then, the uniform exponential decay rate is proved by frequency domain analysis techniques when only one purely elastic undamped rod is present. When the network involves more than one purely elastic undamped rod the lack of exponential decay is proved and nearly sharp polynomial decay rates are deduced under suitable irrationality conditions on the lengths of the rods, based on Diophantine approximation arguments. More general slow decay rates are also derived. Finally, we present some numerical simulations supporting the analytical results.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Asymptotic behavior; Elasticity; Exponential decay; Irrationality conditions; Polynomial decay; Resolvent estimates; Rods; Star-shaped network; Thermoelasticity

Deep Neural Networks: Multi-Classification and Universal Approximation - Hernández Salinas, Martin; Zuazua Iriondo, Enrique (2025)
We demonstrate that a ReLU deep neural network with a width of 2 and a depth of 2N+4M−1 layers can achieve finite sample memorization for any dataset comprising N elements in Rd, where d≥1, and M classes, thereby ensuring accurate classification. By modeling the neural network as a time-discrete nonlinear dynamical system, we interpret the memorization property as a problem of simultaneous or ensemble controllability. This problem is addressed by constructing the network parameters inductively and explicitly, bypassing the need for training or solving any optimization problem. Additionally, we establish that such a network can achieve universal approximation in Lp(Ω;R+), where Ω is a bounded subset of Rd and p∈[1,∞), using a ReLU deep neural network with a width of d+1. We also provide depth estimates for approximating W1,p functions and width estimates for approximating Lp(Ω;Rm) for m≥1. Our proofs are constructive, offering explicit values for the biases and weights involved.<br
https://cris.fau.de/converis/publicweb/Publication/329303154
Keywords: Deep neuronal network; Finite sample memorization; Simultaneous controllability; Nonlinear discrete dynamics; Universal approximation theorem

Differentiability with respect to the initial condition for Hamilton-Jacobi equations - Esteve-Yague, Carlos; Zuazua Iriondo, Enrique (SIAM Journal on Mathematical Analysis, 2022)
We prove that the viscosity solution to a Hamilton-Jacobi equation with a smooth convex Hamiltonian of the form [katex]H(x,p)[/katex] is differentiable with respect to the initial condition. Moreover, the directional Gâteaux derivatives can be explicitly computed almost everywhere in [katex]R^N[/katex] by means of the optimality system of the associated optimal control problem. We also prove that these directional Gâteaux derivatives actually correspond to the unique duality solution to the linear transport equation with discontinuous coefficient, resulting from the linearization of the Hamilton-Jacobi equation. The motivation behind these differentiability results arises from the following optimal inverse-design problem: given a time horizon [katex]T>0[/katex] and a target function [katex]u_T[/katex], construct an initial condition such that the corresponding viscosity solution at time [katex]T[/katex] minimizes the [katex]L^2[/katex]-distance to [katex]u_T[/katex]. Our differentiability results allow us to derive a necessary first-order optimality condition for this optimization problem, and the implementation of gradient-based methods to numerically approximate the optimal inverse design.
Keywords: Hamilton-Jacobi equation; Gâteaux derivatives; inverse design problem; transport equation; duality solutions

Differentiability With Respect To The Initial Condition For Hamilton-jacobi Equations - Esteve-Yague, Carlos; Zuazua Iriondo, Enrique (SIAM Journal on Mathematical Analysis, 2022)
We prove that the viscosity solution to a Hamilton-Jacobi equation with a smooth convex Hamiltonian of the form H(x, p) is differentiable with respect to the initial condition. More-over, the directional Gateaux derivatives can be explicitly computed almost everywhere in R-N by means of the optimality system of the associated optimal control problem. We also prove that, in the one-dimensional case in space and in the quadratic case in any space dimension, these directional Gateaux derivatives actually correspond to the unique duality solution to the linear transport equation with discontinuous coefficient, resulting from the linearization of the Hamilton-Jacobi equation. The motivation behind these differentiability results arises from the following optimal inverse-design problem: given a time horizon T > 0 and a target function uT, construct an initial condition such that the corresponding viscosity solution at time T minimizes the L-2-distance to u(T). Our differentiability results allow us to derive a necessary first-order optimality condition for this optimization problem and the implementation of gradient-based methods to numerically approximate the optimal inverse design.
CRIS-Team WoS Importer:2022-10-14

Dispersion for 1-D Schrödinger and wave equations with BV coefficients - Beli, Constantin N.; Ignat, Liviu I.; Zuazua, Enrique (Annales de l'Institut Henri Poincaré - Analyse Non Linéaire, 2016)
In this paper we analyze the dispersion for one dimensional wave and Schrödinger equations with BV coefficients. In the case of the wave equation we give a complete answer in terms of the variation of the logarithm of the coefficient showing that dispersion occurs if this variation is small enough but it may fail when the variation goes beyond a sharp threshold. For the Schrödinger equation we prove that the dispersion holds under the same smallness assumption on the variation of the coefficient. But, whether dispersion may fail for larger coefficients is unknown for the Schrödinger equation.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Almost periodic functions; BV coefficients; Dispersion and Strichartz estimates; One space dimension; Schrödinger equation; Wave equation

Dynamics and control for multi-agent networked systems: A finite-difference approach - Biccari, Umberto; Ko, Dongnam; Zuazua, Enrique (Mathematical Models & Methods in Applied Sciences, 2019)
We analyze the dynamics of multi-agent collective behavior models and its control theoretical properties. We first derive a large population limit to parabolic diffusive equations. We also show that the nonlocal transport equations commonly derived as the mean-field limit, are subordinated to the first one. In other words, the solution of the nonlocal transport model can be obtained by a suitable averaging of the diffusive one. We then address the control problem in the linear setting, linking the multi-agent model with the spatial semi-discretization of parabolic equations. This allows us to use the existing techniques for parabolic control problems in the present setting and derive explicit estimates on the cost of controlling these systems as the number of agents tends to infinity. We obtain precise estimates on the time of control and the size of the controls needed to drive the system to consensus, depending on the size of the population considered. Our approach, inspired on the existing results for parabolic equations, possibly of fractional type, and in several space dimensions, shows that the formation of consensus may be understood in terms of the underlying diffusion process described by the heat semi-group. In this way, we are able to give precise estimates on the cost of controllability for these systems as the number of agents increases, both in what concerns the needed control time horizon and the size of the controls.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Collective dynamics; consensus; controllability; nonlocal diffusion equations; semi-discretization of PDEs

Eigenvalue bounds for the Gramian operator of the heat equation - Lazar, Martin; Zuazua Iriondo, Enrique (Automatica, 2024)
This paper is concerned with the eigenvalue decay of solution operators to operator Lyapunov equations, a relevant topic in the context of model reduction for parabolic control problems. We mainly focus on the Gramian operator, which arises in the context of control and observation of heat processes in infinite time, which is normally the first step towards observations in a finite time horizon. By improving existing energy and observability estimates for parabolic equations, we obtain both upper and lower bounds on the convergence rate of the eigenvalues of the Gramian operator towards zero. Both bounds follow the same polynomial decay rate, up to a multiplicative constant, which ensures their optimality. This confirms the slow decay of the eigenvalues and limits the efficiency of model reduction. The theoretical findings are supported by numerical results

Exact Controllability and Stabilization of the Wave Equation - Zuazua Iriondo, Enrique (Springer Nature, 2024)
This comprehensive monograph illustrates the intricate realm of controllability and stabilization of wave phenomena. Authored by an expert in the field, this book integrates J. L. Lion's renowned HUM method, multiplier techniques, and the construction of Lyapunov functionals. Through meticulous analysis and practical applications, this book provides invaluable insights for researchers seeking to navigate the expansive domain of wave-like equations and their control. Whether you are a seasoned mathematician or a newcomer to the field, this book serves as an indispensable guide, offering a thorough introduction and essential tools for understanding and controlling wave phenomena. Typesettings and translation: Eng. Darlis Bracho Tudares www.dcn.nat.fau.eu/darlis-bracho-tudares
CRIS-Team Scopus Importer:2024-11-08
Keywords: Control; Energy estimates; Lyapunov functionals; Stabilization; Wave equation

Exact controllability of the semi-linear wave equation - Zhang, Xu; Zuazua, Enrique (Unsolved Problems in Mathematical Systems and Control Theory, 2009)
CRIS-Team Scopus Importer:2021-06-15

Exact Controllability of the Time Discrete Wave Equation: A Multiplier Approach - Zhang, Xu; Zheng, Chuang; Zuazua, Enrique (Applied and Numerical Partial Differential Equations, 2010)
In this paper we summarize our recent results on the exact boundary controllability of a trapezoidal time discrete wave equation in a bounded domain. It is shown that the projection of the solution in an appropriate space in which the high frequencies have been filtered is exactly controllable with uniformly bounded controls (with respect to the time-step). By classical duality arguments, the problem is reduced to a boundary observability inequality for a time-discrete wave equation. Using multiplier techniques the uniform observability property is proved in a class of filtered initial data. The optimality of the filtering parameter is also analyzed.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Exact controllability; filtering.; multiplier technique; observability; time discretization; wave equation

Exact penalization of terminal constraints for optimal control problems - Gugat, Martin; Zuazua, Enrique (Optimal Control Applications & Methods, 2016)
We study optimal control problems for linear systems with prescribed initial and terminal states. We analyze the exact penalization of the terminal constraints. We show that for systems that are exactly controllable, the norm-minimal exact control can be computed as the solution of an optimization problem without terminal constraint but with a nonsmooth penalization of the end conditions in the objective function, if the penalty parameter is sufficiently large. We describe the application of the method for hyperbolic and parabolic systems of partial differential equations, considering the wave and heat equations as particular examples. Copyright © 2016 John Wiley & Sons, Ltd.
Keywords: exact controllability; optimal control; terminal constraint; exact penalization; wave equation; heat equation; moment equations; method of moments; L1 optimal control; abstract Cauchy problems; nonsmooth optimization

Exact Sequence Classification with Hardmax Transformers - Alcalde Zafra, Albert; Fantuzzi, Giovanni; Zuazua Iriondo, Enrique (2025)
We prove that hardmax attention transformers perfectly classify datasets of N labeled sequences in R^d, d≥ 2. Specifically, given N sequences with an arbitrary but finite length in R^d, we construct a transformer with O(N) blocks and O(Nd) parameters perfectly classifying this dataset. Our construction achieves the best complexity estimate to date, independent of the length of the sequences, by innovatively alternating feed-forward and self-attention layers and by capitalizing on the clustering effect inherent to the latter. Our novel constructive method also uses low-rank parameter matrices within the attention mechanism, a common practice in real-life transformer implementations. Consequently, our analysis holds twofold significance: it substantially advances the mathematical theory of transformers and it rigorously justifies their exceptional real-world performance in sequence classification task
https://cris.fau.de/converis/publicweb/Publication/334528458
Keywords: Transformers; Self-Attention Mechanism; Exact Classification; Theory of Deep Learning

Exponential convergence to steady-states for trajectories of a damped dynamical system modeling adhesive strings - Coclite, Giuseppe Maria; De Nitti, Nicola; Maddalena, Francesco; Orlando, Gianluca; Zuazua Iriondo, Enrique (Mathematical Models & Methods in Applied Sciences, 2024)
We study the global well-posedness and asymptotic behavior for a semilinear damped wave equation with Neumann boundary conditions, modeling a one-dimensional linearly elastic body interacting with a rigid substrate through an adhesive material. The key feature of of the problem is that the interplay between the nonlinear force and the boundary conditions allows for a continuous set of equilibrium points. We prove an exponential rate of convergence for the solution towards a (uniquely determined) equilibrium point.
CRIS-Team Scopus Importer:2024-05-03
Keywords: adhesion phenomena; Damped wave equation; decay; long-time asymptotics

Exponential convergence to steady-states for trajectories of a damped dynamical system modelling adhesive strings - Coclite, Giuseppe Maria; De Nitti, Nicola; Maddalena, Francesco; Orlando, Gianluca; Zuazua Iriondo, Enrique (2024)
We study the global well-posedness and asymptotic behavior for a semilinear damped wave equation with Neumann boundary conditions, modelling a one-dimensional linearly elastic body interacting with a rigid substrate through an adhesive material. The key feature of of the problem is that the interplay between the nonlinear force and the boundary conditions allows for a continuous set of equilibrium points. We prove an exponential rate of convergence for the solution towards a (uniquely determined) equilibrium point.
https://cris.fau.de/converis/publicweb/Publication/313437958
Keywords: Damped wave equation; adhesion phenomena; long-time asymptotics; decay

FedADMM-InSa: An Inexact and Self-Adaptive ADMM for Federated Learning - Song, Yongcun; Wang, Ziqi; Zuazua Iriondo, Enrique (Neural Networks, 2025)
Federated learning (FL) is a promising framework for learning from distributed data while maintaining privacy. The development of efficient FL algorithms encounters various challenges, including heterogeneous data and systems, limited communication capacities, and constrained local computational resources. Recently developed FedADMM methods show great resilience to both data and system heterogeneity. However, they still suffer from performance deterioration if the hyperparameters are not carefully tuned. To address this issue, we propose an inexact and self-adaptive FedADMM algorithm, termed FedADMM-InSa. First, we design an inexactness criterion for the clients' local updates to eliminate the need for empirically setting the local training accuracy. This inexactness criterion can be assessed by each client independently based on its unique condition, thereby reducing the local computational cost and mitigating the undesirable straggle effect. The convergence of the resulting inexact ADMM is proved under the assumption of strongly convex loss functions. Additionally, we present a self-adaptive scheme that dynamically adjusts each client's penalty parameter, enhancing algorithm robustness by mitigating the need for empirical penalty parameter choices for each client. Extensive numerical experiments on both synthetic and real-world datasets have been conducted. As validated by some tests, our FedADMM-InSa algorithm improves model accuracy by 7.8% while reducing clients' local workloads by 55.7% compared to benchmark algorithm
CRIS-Team Scopus Importer:2024-10-25
Keywords: Machine Learning; Federated learning; Distributed; Parallel; and Cluster Computing; Optimization and Control

Filtered gradient algorithms for inverse design problems of one-dimensional burgers equation - Gosse, Laurent; Zuazua, Enrique (Innovative Algorithms and Analysis, 2017)
Inverse design for hyperbolic conservation laws is exemplified through the 1D Burgers equation which is motivated by aircraft’s sonic-boom minimization issues. In particular, we prove that, as soon as the target function (usually a Nwave) isn’t continuous, there is a whole convex set of possible initial data, the backward entropy solution being possibly its centroid. Further, an iterative strategy based on a gradient algorithm involving “reversible solutions” solving the linear adjoint problem is set up. In order to be able to recover initial profiles different from the backward entropy solution, a filtering step of the backward adjoint solution is inserted, mostly relying on scale-limited (wavelet) subspaces. Numerical illustrations, along with profiles similar to F-functions, are presented.
CRIS-Team Scopus Importer:2021-06-15

Finite Element Approximation of the Hardy Constant - Della Pietra, Francesco; Fantuzzi, Giovanni; Ignat, Liviu I.; Masiello, Alba Lia; Paoli, Gloria; Zuazua Iriondo, Enrique (Journal of Convex Analysis, 2024)
We consider finite element approximations to the optimal constant for the Hardy inequality with exponent p = 2 in bounded domains of dimension n = 1 or n ≥ 3. For finite element spaces of piecewise linear and continuous functions on a mesh of size h, we prove that the approximate Hardy constant converges to the optimal Hardy constant at a rate proportional to 1/|log h|2. This result holds in dimension n = 1, in any dimension n ≥ 3 if the domain is the unit ball and the finite element discretization exploits the rotational symmetry of the problem, and in dimension n = 3 for general finite element discretizations of the unit ball. In the first two cases, our estimates show excellent quantitative agreement with values of the discrete Hardy constant obtained computationally.
CRIS-Team Scopus Importer:2024-04-12
Keywords: finite element method; Hardy constant; Hardy inequality

Finite element approximation of the Hardy constant - Della Pietra, Francesco; Fantuzzi, Giovanni; Ignat, Liviu I.; Masiello, Alba Lia; Paoli, Gloria; Zuazua Iriondo, Enrique (2024)
We consider finite element approximations to the optimal constant for the Hardy inequality with exponent p=2 in bounded domains of dimension n=1 or n≥ 3. For finite element spaces of piecewise linear and continuous functions on a mesh of size h, we prove that the approximate Hardy constant, S_h^n, converges to the optimal Hardy constant S^n no slower than O(1/h). We also show that the convergence is no faster than O(1/h^2) if n=1 or if n≥ 3, the domain is the unit ball, and the finite element discretization exploits the rotational symmetry of the problem. Our estimates are compared to exact values for S_h^n obtained computationally.<br
https://cris.fau.de/converis/publicweb/Publication/308553439

Flow decomposition for heat equations with memory - Wang, Gengsheng; Zhang, Yubiao; Zuazua Iriondo, Enrique (Journal De Mathematiques Pures Et Appliquees, 2021)
We build up a decomposition for the flow generated by the heat equation with a real analytic memory kernel. It consists of three components: The first one is of parabolic nature; the second one gathers the hyperbolic component of the dynamics, with null velocity of propagation; the last one exhibits a finite smoothing effect. This decomposition reveals the hybrid parabolic-hyperbolic nature of the flow and clearly illustrates the significant impact of the memory term on the parabolic behavior of the system in the absence of memory terms.<br
Keywords: Heat equations with memory; decomposition of the flow; hybrid parabolichyperbolic behavior

FLUX IDENTIFICATION FOR 1-d SCALAR CONSERVATION LAWS IN THE PRESENCE OF SHOCKS - Castro, Carlos; Zuazua, Enrique (Mathematics of Computation, 2011)
We consider the problem of flux identification for 1-d scalar conservation laws formulating it as an optimal control problem. We introduce a new optimization strategy to compute numerical approximations of minimizing fluxes.
CRIS-Team WoS Importer:2019-12-13

Fourier series and sidewise control of 1-d waves - Zuazua Iriondo, Enrique (Documents Mathématiques of the French Mathematical Society (SMF), 2024)
We discuss the sidewise control properties of 1-d waves. In analogy with classical control and inverse problems for wave propagation, the problem consists on controlling the behaviour of waves on part of the boundary of the domain where they propagate, by means of control actions localised on a different subset of the boundary. In contrast with classical problems, the goal is not to control the dynamics of the waves on the interior of the domain, but rather their boundary traces. It is therefore a goal oriented controllability problem. We propose a duality method that reduces the problem to suitable new observability inequalities, which consist of estimating the boundary traces of waves on part of the boundary from boundary measurements done on another subset of the boundary. These inequalities lead to novel questions that do not seem to be treatable by the classical techniques employed in the field, such as Carleman inequalities, non-harmonic Fourier series, microlocal analysis and multipliers. We propose a genuinely 1-d solution method, based on sidewise energy propagation estimates yielding a complete sharp solution. The obtained observability results can be reinterpreted in terms of Fourier series. This leads to new non-standard questions in the context of non-hamonic Fourier series

Fourier series and sidewise profile control of 1-d waves - Zuazua Iriondo, Enrique (2024)
We discuss the sidewise control properties of 1-d waves. In analogy with classical control and inverse problems for wave propagation, the problem consists on controlling the behaviour of waves on part of the boundary of the domain where they propagate, by means of control actions localised on a different subset of the boundary. In contrast with classical problems, the goal is not to control the dynamics of the waves on the interior of the domain, but rather their boundary traces. It is therefore a goal oriented controllability problem. We propose a duality method that reduces the problem to suitable new observability inequalities, which consist of estimating the boundary traces of waves on part of the boundary from boundary measurements done on another subset of the boundary. These inequalities lead to novel questions that do not seem to be treatable by the classical techniques employed in the field, such as Carleman inequalities, non-harmonic Fourier series, microlocal analysis and multipliers. We propose a genuinely 1-d solution method, based on sidewise energy propagation estimates yielding a complete sharp solution. The obtained observability results can be reinterpreted in terms of Fourier series. This leads to new non-standard questions in the context of non-hamonic Fourier series
https://cris.fau.de/converis/publicweb/Publication/308864875

Fourier series and sidewise profile control of 1-d waves - Zuazua Iriondo, Enrique (2024)
We discuss the sidewise control properties of 1-d waves. In analogy with classical control and inverse problems for wave propagation, the problem consists on controlling the behaviour of waves on part of the boundary of the domain where they propagate, by means of control actions localised on a different subset of the boundary. In contrast with classical problems, the goal is not to control the dynamics of the waves on the interior of the domain, but rather their boundary traces. It is therefore a goal oriented controllability problem. We propose a duality method that reduces the problem to suitable new observability inequalities, which consist of estimating the boundary traces of waves on part of the boundary from boundary measurements done on another subset of the boundary. These inequalities lead to novel questions that do not seem to be treatable by the classical techniques employed in the field, such as Carleman inequalities, non-harmonic Fourier series, microlocal analysis and multipliers. We propose a genuinely 1-d solution method, based on sidewise energy propagation estimates yielding a complete sharp solution. The obtained observability results can be reinterpreted in terms of Fourier series. This leads to new non-standard questions in the context of non- hamonic Fourier series.
https://cris.fau.de/converis/publicweb/Publication/308484723

Full probabilistic solution of a finite dimensional linear control system with random initial and final conditions - Cortés, Juan C.; Navarro-Quiles, Ana; Romero, J. V.; Roselló, María Dolores; Zuazua Iriondo, Enrique (Journal of the Franklin Institute-Engineering and Applied Mathematics, 2019)
In this paper we perform a complete probabilistic study of a finite dimensional linear control system with uncertainty. The controllability condition with random initial data and final target is analysed. To conduct this investigation we determine the first probability density function of the control and the solution of the random control problem under different scenarios. To achieve this objective, Random Variable Transformation technique is extensively applied. Several examples illustrate the theoretical results.<br
Keywords: random linear control system; first probability density function; random variable transformation technique

Gaussian Beam Ansatz for Finite Difference Wave Equations - Biccari, Umberto; Zuazua, Enrique (Foundations of Computational Mathematics, 2023)
This work is concerned with the construction of Gaussian Beam (GB) solutions for the numerical approximation of wave equations, semi-discretized in space by finite difference schemes. GB are high-frequency solutions whose propagation can be described, both at the continuous and at the semi-discrete levels, by microlocal tools along the bi-characteristics of the corresponding Hamiltonian. Their dynamics differ in the continuous and the semi-discrete setting, because of the high-frequency gap between the Hamiltonians. In particular, numerical high-frequency solutions can exhibit spurious pathological behaviors, such as lack of propagation in space, contrary to the classical space-time propagation properties of continuous waves. This gap between the behavior of continuous and numerical waves introduces also significant analytical difficulties, since classical GB constructions cannot be immediately extrapolated to the finite difference setting, and need to be properly tailored to accurately detect the propagation properties in discrete media. Our main objective in this paper is to present a general and rigorous construction of the GB ansatz for finite difference wave equations, and corroborate this construction through accurate numerical simulations.
CRIS-Team Scopus Importer:2023-10-27
Keywords: Finite difference; Gaussian Beam ansatz; Wave equation

Gaussian Beam ansatz for finite difference wave equations - Biccari, Umberto; Zuazua Iriondo, Enrique (Foundations of Computational Mathematics, 2025)
This work is concerned with the construction of Gaussian Beam (GB) solutions for the numerical approximation of wave equations, semi-discretized in space by finite difference schemes. GB are high-frequency solutions whose propagation can be described, both at the continuous and at the semi-discrete levels, by microlocal tools along the bi-characteristics of the corresponding Hamiltonian. Their dynamics differ in the continuous and the semi-discrete setting, because of the high-frequency gap between the Hamiltonians. In particular, numerical high-frequency solutions can exhibit spurious pathological behaviors, such as lack of propagation in space, contrary to the classical space-time propagation properties of continuous waves. This gap between the behavior of continuous and numerical waves introduces also significant analytical difficulties, since classical GB constructions cannot be immediately extrapolated to the finite difference setting, and need to be properly tailored to accurately detect the propagation properties in discrete media. Our main objective in this paper is to present a general and rigorous construction of the GB ansatz for finite difference wave equations, and corroborate this construction through accurate numerical simulations.
Keywords: Wave equation; Finite difference; Gaussian Beam ansatz

Generation of 2D water waves by moving bottom disturbances - Nersisyan, Hayk; Dutykh, Denys; Zuazua, Enrique (IMA Journal of Applied Mathematics, 2014)
We investigate the potential and limitations of the wave generation by disturbances moving at the bottom. More precisely, we assume that the wavemaker is composed of an underwater object of a given shape which can be displaced according to a prescribed trajectory. We address the practical question of computing the wavemaker shape and trajectory generating a wave with prescribed characteristics. For the sake of simplicity, we model the hydrodynamics by a generalized forced Benjamin-Bona-Mahony equation. This practical problem is reformulated as a constrained non-linear optimization problem. Additional constraints are imposed in order to fulfil various practical design requirements. Finally, we present some numerical results in order to demonstrate the feasibility and performance of the proposed methodology.
CRIS-Team Scopus Importer:2021-06-15
Keywords: BBM equation; moving bottom; optimization; wave generation

Greedy controllability of finite dimensional linear systems - Lazar, Martin; Zuazua, Enrique (Automatica, 2016)
We analyse the problem of controllability for parameter dependent linear finite-dimensional systems. The goal is to identify the most distinguished realisations of those parameters so to better describe or approximate the whole range of controls. We adapt recent results on greedy and weak greedy algorithms for parameter dependent PDEs or, more generally, abstract equations in Banach spaces. Our results lead to optimal approximation procedures that, in particular, perform better than simply sampling the parameter-space to compute the controls for each of the parameter values. We apply these results for the approximate control of finite-difference approximations of the heat and the wave equation. The numerical experiments confirm the efficiency of the methods and show that the number of weak-greedy samplings that are required is particularly low when dealing with heat-like equations, because of the intrinsic dissipativity that the model introduces for high frequencies.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Finite-differences; Greedy control; Heat equation; Parametrised ODEs and PDEs; Wave equation; Weak-greedy

Greedy optimal control for elliptic problems and its application to turnpike problems - Hernandez-Santamaria, Victor; Lazar, Martin; Zuazua, Enrique (Numerische Mathematik, 2019)
We adapt and apply greedy methods to approximate in an efficient way the optimal controls for parameterized elliptic control problems. Our results yield an optimal approximation procedure that, in particular, performs better than simply sampling the parameter-space to compute controls for each parameter value. The same method can be adapted for parabolic control problems, but this leads to greedy selections of the realizations of the parameters that depend on the initial datum under consideration. The turnpike property (which ensures that parabolic optimal control problems behave nearly in a static manner when the control horizon is long enough) allows using the elliptic greedy choice of the parameters in the parabolic setting too. We present various numerical experiments and an extensive discussion of the efficiency of our methodology for parabolic control and indicate a number of open problems arising when analyzing the convergence of the proposed algorithms.
CRIS-Team Scopus Importer:2021-06-15

Greedy search of optimal approximate solutions - Lazar, Martin; Zuazua Iriondo, Enrique (Pure and Applied Functional Analysis, 2023)
In this paper we develop a procedure to deal with a family of parameter-dependent ill-posed problems, for which the exact solution in general does not exist. The original problems are relaxed by considering corresponding approximate ones, whose optimal solutions are well defined, where the optimality is determined by the minimal norm requirement. The procedure is based upon greedy algorithms that preserve, at least asymptotically, Kolmogorov approximation rates. In order to provide a-priori estimates for the algorithm, a Tychonoff-type regularization is applied, which adds an additional parameter to the model. The theory is developed in an abstract theoretical framework that allows its application to different kinds of problems. We present a specific example that considers a family of ill-posed elliptic problems. The required general assumptions in this case translate to rather natural uniform lower and upper bounds on coefficients of the considered operators.<br
Keywords: Optimal approximate solutions; parametrized PDEs; greedy algorithms

HARDY INEQUALITIES, OBSERVABILITY, AND CONTROL FOR THE WAVE AND SCHRODINGER EQUATIONS WITH SINGULAR POTENTIALS - Vancostenoble, J.; Zuazua, Enrique (SIAM Journal on Mathematical Analysis, 2009)
We address the question of exact controllability of the wave and Schrodinger equations perturbed by a singular inverse-square potential. Exact boundary controllability is proved in the range of subcritical coefficients of the singular potential and under suitable geometric conditions. The proof relies on the method of multipliers. The key point in the proof of the observability inequality is a suitable Hardy-type inequality with sharp constants. On the contrary, in the supercritical case, we prove that exact controllability is false.
CRIS-Team WoS Importer:2019-12-13

High frequency wave packets for the Schrodinger equation and its numerical approximations - Marica, Aurora; Zuazua, Enrique (Comptes Rendus Mathematique, 2011)
We build Gaussian wave packets for the linear Schrodinger equation and its finite difference space semi-discretization and illustrate the lack of uniform dispersive properties of the numerical solutions as established in Ignat and Zuazua (2009) [6]. It is by now well known that bigrid algorithms provide filtering mechanisms allowing to recover the uniformity of the dispersive properties as the mesh size goes to zero. We analyze and illustrate numerically how these high frequency wave packets split and propagate under these bigrid filtering mechanisms, depending on how the fine grid/coarse grid filtering is implemented. (C) 2010 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.
CRIS-Team WoS Importer:2019-12-13

IDENTIFICATION OF THE CLASS OF INITIAL DATA FOR THE INSENSITIZING CONTROL OF THE HEAT EQUATION - De Teresa, Luz; Zuazua, Enrique (Communications on Pure and Applied Analysis, 2009)
This paper is devoted to analyze the class of initial data that can be insensitized for the heat equation. This issue has been extensively addressed in the literature both in the case of complete and approximate insensitization (see [19] and [1], respectively).
CRIS-Team WoS Importer:2019-12-13

Improved multipolar hardy inequalities - Cazacu, Cristian; Zuazua, Enrique (Studies in Phase Space Analysis with Applications to PDEs, 2013)
In this paper we prove optimal Hardy-type inequalities for Schrödinger operators with positive multi-singular inverse square potentials of the form (Formula presented.) More precisely, we show that A λ is nonnegative in the sense of L 2 quadratic forms in RN, if and only if λ≤(N-2)2/n2, independently of the number n and location of the singularities xiϵRN, where N ≥ 3 denotes the space dimension. This aims to complement some of the results in Bosi et al. (Comm. Pure Appl. Anal. 7:533–562, 2008) obtained by the “expansion of the square” method. Due to the interaction of poles, our optimal result provides a singular quadratic potential behaving like (n−1)(N −2)2/(n2|x−xi|2) at each pole xi. Besides, the authors in Bosi et al. (Comm. Pure Appl. Anal. 7:533–562, 2008) showed optimal Hardy inequalities for Schrödinger operators with a finite number of singular poles of the type Bλ:=-Δ-Σi=1 nλ/|x-xi|2, up to lower order L 2-reminder terms. By means of the optimal results obtained for A λ, we also build some examples of bounded domains Ω in which these lower order terms can be removed in H 0 1(Ω). In this way we obtain new lower bounds for the optimal constant in the standard multi-singular Hardy inequality for the operator B λ in bounded domains. The best lower bounds are obtained when the singularities xi are located on the boundary of the domain.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Hardy inequalities; Multipolar potentials; Schrödinger operators

Initial data identification for the one-dimensional Burgers equation - Liard, Thibault; Zuazua Iriondo, Enrique (IEEE Transactions on Automatic Control, 2022)
In this paper, we study the problem of identification for the one-dimensional Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the property of non-backward uniqueness of Burgers equation, there may exist multiple initial data leading to the same given target. In [13], [17], the authors fully characterize the set of initial data leading to a given target using the classical Lax-Hopf formula. In this note, an alternative proof based only on generalized backward characteristics is given. This leads to the hope of investigate systems of conservation laws in one dimension where the classical Lax-Hopf formula doesn’t hold anymore. Moreover, numerical illustrations are presented using as a target, a function optimized for minimum pressure rise in the context of sonic-boom minimization problems. All of initial data leading to this given target are constructed using a wave-front tracking algorithm.<br
Keywords: Entropy; Minimization; Mathematical model; Airplanes; Target tracking; Shape; Optimal control

Internal Controllability for Parabolic Systems Involving Analytic Non-local Terms - Lissy, Pierre; Zuazua, Enrique (Chinese Annals of Mathematics Series B, 2018)
This paper deals with the problem of internal controllability of a system of heat equations posed on a bounded domain with Dirichlet boundary conditions and perturbed with analytic non-local coupling terms. Each component of the system may be controlled in a different subdomain. Assuming that the unperturbed system is controllable—a property that has been recently characterized in terms of a Kalman-like rank condition—the authors give a necessary and sufficient condition for the controllability of the coupled system under the form of a unique continuation property for the corresponding elliptic eigenvalue system. The proof relies on a compactness-uniqueness argument, which is quite unusual in the context of parabolic systems, previously developed for scalar parabolic equations. The general result is illustrated by two simple examples.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Analyticity; Kalman rank condition; Non-local potentials; Null controllability; Parabolic systems; Spectral unique continuation

Internal observability for coupled systems of linear partial differential equations - Lissy, Pierre; Zuazua, Enrique (SIAM Journal on Control and Optimization, 2019)
We deal with the internal observability for some coupled systems of partial differential equations with constant or time-dependent coupling terms by means of a reduced number of observed components. We prove new general observability inequalities under some Kalman-like or Silverman-Meadows-like condition. Our proofs combine the observability properties of the underlying scalar equation with algebraic manipulations. In the more specific case of systems of heat equations with constant coefficients and nondiagonalizable diffusion matrices, we also give a new necessary and sufficient condition for observability in the natural L2-setting. The proof relies on the use of the Lebeau-Robbiano strategy together with a precise study of the cost of controllability for linear ordinary differential equations, and allows us to treat the case where each component of the system is observed in a different subdomain.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Observability inequalities; Partial differential equations; Rank conditions; Systems

Interplay between depth and width for interpolation in neural ODEs - Álvarez-López, Antonio; Slimane, Arselane Hadj; Zuazua Iriondo, Enrique (Neural Networks, 2024)
Neural ordinary differential equations have emerged as a natural tool for supervised learning from a control perspective, yet a complete understanding of the role played by their architecture remains elusive. In this work, we examine the interplay between the width p and the number of transitions between layers L (corresponding to a depth of L+1). Specifically, we construct explicit controls interpolating either a finite dataset D, comprising N pairs of points in Rd, or two probability measures within a Wasserstein error margin ɛ>0. Our findings reveal a balancing trade-off between p and L, with L scaling as 1+O(N/p) for data interpolation, and as 1+Op−1+(1+p)−1ɛ−d for measures. In the high-dimensional and wide setting where d,p>N, our result can be refined to achieve L=0. This naturally raises the problem of data interpolation in the autonomous regime, characterized by L=0. We adopt two alternative approaches: either controlling in a probabilistic sense, or by relaxing the target condition. In the first case, when p=N we develop an inductive control strategy based on a separability assumption whose probability increases with d. In the second one, we establish an explicit error decay rate with respect to p which results from applying a universal approximation theorem to a custom-built Lipschitz vector field interpolating D.
CRIS-Team Scopus Importer:2024-09-13
Keywords: Depth; Neural ODEs; Simultaneous controllability; Transport control; Wasserstein distance; Width

Interpolation and approximation via Momentum ResNets and Neural ODEs - Ruiz-Balet, Domènec; Affili, Elisa; Zuazua Iriondo, Enrique (IEEE Control Systems Letters, 2022)
In this article, we explore the effects of memory terms in continuous-layer Deep Residual Networks by studying Neural ODEs (NODEs). We investigate two types of models. On one side, we consider the case of Residual Neural Networks with dependence on multiple layers, more precisely Momentum ResNets. On the other side, we analyse a Neural ODE with auxiliary states playing the role of memory states. We examine the interpolation and universal approximation properties for both architectures through a simultaneous control perspective. We also prove the ability of the second model to represent sophisticated maps, such as parametrizations of time-dependent functions. Numerical simulations complement our study.<br
Keywords: Memory models; NODEs; Momentum ResNets; Simultaneous Control; Simultaneous tracking controllability; Universal Approximation; Deep Learning

Inverse design for the one-dimensional Burgers equation - Liard, Thibault; Zuazua Iriondo, Enrique (2019)
In this paper, we study the problem of inverse design for the one-dimensional Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the time-irreversibility of the Burgers equation, some target functions are unattainable from solutions of this equation, making the inverse problem under consideration ill-posed. To get around this issue, we introduce an optimal control problem which consists in minimizing the difference between the predictions of the Burgers equation and the observations of the system at a final time in L^2(R) norm. The two main contributions of this work are the following: We fully characterize the set of minimizers of the aforementioned optimal control problem A wave-front tracking method is implemented to construct numerically all of them One of minimizers is the backward entropy solution, constructed using a backward-forward metho
Keywords: Inverse problems; Conservation Laws; Entropy solutions; Backward-Forward approach; Optimal Control Problem; Wave-front tracking algorithm.

Large-time asymptotics for hyperbolic systems with non-symmetric relaxation: An algorithmic approach - Crin-Barat, Timothée; Liverani, Lorenzo; Shou, Ling-yun; Zuazua Iriondo, Enrique (2025)
We study the stability of one-dimensional linear hyperbolic systems with non-symmetric relaxation. Introducing a new frequency-dependent Kalman stability condition, we prove an abstract decay result underpinning a form of inhomogeneous hypocoercivity. In contrast with the homogeneous setting, the decay rates depend on how the Kalman condition is fulfilled and, in most cases, a loss of derivative occurs: one must assume an additional regularity assumption on the initial data to ensure the decay. Under structural assumptions, we refine our abstract result by providing an algorithm, of wide applicability, for the construction of Lyapunov functionals. This allows us to systematically establish decay estimates for a given system and uncover algebraic cancellations (beyond the reach of the Kalman-based approach) reducing the loss of derivatives in high frequencies. To demonstrate the applicability of our method, we derive new stability results for the Sugimoto model, which describes the propagation of nonlinear acoustic waves, and for a beam model of Timoshenko type with memor
https://cris.fau.de/converis/publicweb/Publication/336363443

Large Time Asymptotics for Partially Dissipative Hyperbolic Systems - Beauchard, Karine; Zuazua, Enrique (Archive for Rational Mechanics and Analysis, 2011)
This work is concerned with (n-component) hyperbolic systems of balance laws in m space dimensions. First, we consider linear systems with constant coefficients and analyze the possible behavior of solutions as t -> infinity. Using the Fourier transform, we examine the role that control theoretical tools, such as the classical Kalman rank condition, play. We build Lyapunov functionals allowing us to establish explicit decay rates depending on the frequency variable. In this way we extend the previous analysis by Shizuta and Kawashima under the so-called algebraic condition (SK). In particular, we show the existence of systems exhibiting more complex behavior than the one that the (SK) condition allows. We also discuss links between this analysis and previous literature in the context of damped wave equations, hypoellipticity and hypocoercivity. To conclude, we analyze the existence of global solutions around constant equilibria for nonlinear systems of balance laws. Our analysis of the linear case allows proving existence results in situations that the previously existing theory does not cover.
CRIS-Team WoS Importer:2019-12-13

Large time asymptotics for partially dissipative hyperbolic systems without Fourier analysis: application to the nonlinearly damped p-system - Crin-Barat, Timothée; Zuazua Iriondo, Enrique; et al. (2025)
A new framework to obtain time-decay estimates for partially dissipative hyperbolic systems set on the real line is developed. Under the classical Shizuta-Kawashima (SK) stability condition, equivalent to the Kalman rank condition in control theory, the solutions of these systems decay exponentially in time for high frequencies and polynomially for low ones. This allows to derive a sharp description of the space-time decay of solutions for large time. However, such analysis relies heavily on the use of the Fourier transform that we avoid here, developing the "physical space version" of the hyperbolic hypocoercivity approach introduced by Beauchard and Zuazua, to prove new asymptotic results in the linear and nonlinear settings. The new physical space version of the hyperbolic hypocoercivity approach allows to recover the natural heat-like time-decay of solutions under sharp rank conditions, without employing Fourier analysis or L1 assumptions on the initial data. Taking advantage of this Fourier-free framework, we establish new enhanced time-decay estimates for initial data belonging to weighted Sobolev spaces. These results are then applied to the nonlinear compressible Euler equations with linear damping. We also prove the logarithmic stability of the nonlinearly damped p-syste
https://cris.fau.de/converis/publicweb/Publication/310644407

Large-time asymptotics in deep learning - Esteve Yagüe, Carlos; Geshkovski, Borjan; Pighin, Dario; Zuazua Iriondo, Enrique (2024)
We consider the neural ODE perspective of supervised learning and study the impact of the final time T (which may indicate the depth of a corresponding ResNet) in training. For the classical L2–regularized empirical risk minimization problem, whenever the neural ODE dynamics are homogeneous with respect to the parameters, we show that the training error is at most of the order (1T). Furthermore, if the loss inducing the empirical risk attains its minimum, the optimal parameters converge to minimal L2–norm parameters which interpolate the dataset. By a natural scaling between T and the regularization hyperparameter λ we obtain the same results when λ↘0 and T is fixed. This allows us to stipulate generalization properties in the overparametrized regime, now seen from the large depth, neural ODE perspective. To enhance the polynomial decay, inspired by turnpike theory in optimal control, we propose a learning problem with an additional integral regularization term of the neural ODE trajectory over [0,T]. In the setting of ℓp–distance losses, we prove that both the training error and the optimal parameters are at most of the order (e−μt) in any t∈[0,T]. The aforementioned stability estimates are also shown for continuous space-time neural networks, taking the form of nonlinear integro-differential equations. By using a time-dependent moving grid for discretizing the spatial variable, we demonstrate that these equations provide a framework for addressing ResNets with variable widths.<br
https://cris.fau.de/converis/publicweb/Publication/317343122
Keywords: Deep learning; ResNets; neural ODEs; regularization path; optimal control; exponential stability; approximation; turnpike theory

Large-time asymptotics, vanishing viscosity and numerics for 1-D scalar conservation laws - Ignat, Liviu I.; Pozo, Alejandro; Zuazua, Enrique (Mathematics of Computation, 2015)
In this paper we analyze the large time asymptotic behavior of the discrete solutions of numerical approximation schemes for scalar hyperbolic conservation laws. We consider three monotone conservative schemes that are consistent with the one-sided Lipschitz condition (OSLC): Lax-Friedrichs, Engquist-Osher and Godunov. We mainly focus on the inviscid Burgers equation, for which we know that the large time behavior is of a self-similar nature, described by a two-parameter family of N-waves. We prove that, at the numerical level, the large time dynamics depends on the amount of numerical viscosity introduced by the scheme: while Engquist-Osher and Godunov yield the same N-wave asymptotic behavior, the Lax-Friedrichs scheme leads to viscous self-similar profiles, corresponding to the asymptotic behavior of the solutions of the continuous viscous Burgers equation. The same problem is analyzed in the context of self-similar variables that lead to a better numerical performance but to the same dichotomy on the asymptotic behavior: N-waves versus viscous waves. We also give some hints to extend the results to more general fluxes. Some numerical experiments illustrating the accuracy of the results of the paper are also presented.
CRIS-Team Scopus Importer:2021-06-15

Large time control and turnpike properties for wave equations - Zuazua, Enrique (Annual Reviews in Control, 2017)
In the last decades mathematical control theory has been extensively developed to handle various models, including Ordinary and Partial Differential Equations (ODE and PDE), both of deterministic and stochastic nature, discrete and hybrid systems. However, little attention has been paid to the length of the time horizon of control, which is necessarily long in many applications, and to how it affects the nature of controls and controlled trajectories. The turnpike property refers precisely to those aspects and stresses the fact that, often, optimal controls and trajectories, in long time intervals, undergo some relevant asymptotic simplification property ensuring that, during most of the time-horizon of control, optimal pairs remain close to the steady-state optimal one. Due to the intrinsic finite velocity of propagation and the oscillatory nature of solutions of the free wave equation, optimal controls for waves are typically of oscillatory nature. But, despite this, as we shall see, under suitable coercivity conditions on the cost functional to be minimised and when controllability holds, the turnpike property is also fulfilled for the wave equation. When this occurs, the approximation of the time-depending control problem by the steady-state one is justified, a fact that is often employed in applications to reduce the computational cost. We present some recent results of this nature for the wave equation and other closely related conservative systems, and discuss some other related issues and a number of relevant open problems that arise in this field.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Controllability; Long time horizons; Optimal control; The turnpike property; Waves

Linear projection-based CEST parameter estimation - Glang, Felix; Zaiss, Moritz; Laun, Frederik Bernd; Fabian, Moritz S.; German, Alexander; Khakzar, Katrin; Mennecke, Angelika; Liebert, Andrzej; Herz, Kai; Liebig, Patrick; Kasper, Burkhard S.; Schmidt, Manuel; et al. (NMR in Biomedicine, 2022)
Isolated evaluation of multi-parametric in vivo CEST MRI often requires complex computational processing for both correction of B0 and B1 inhomogeneity and contrast generation. For that, sufficiently densely sampled Z-spectra need to be acquired. The list of acquired frequency offsets largely determines the total CEST acquisition time, while potentially representing redundant information. In this work, a linear projection-based multi parametric CEST evaluation method is introduced that offers fast B0 and B1 inhomogeneity correction, contrast generation and feature selection for CEST data, enabling reduction of the overall measurement time. To that end, CEST data acquired at 7T in 6 healthy subjects and in one brain tumor patient were conventionally evaluated by interpolation-based inhomogeneity correction and Lorentzian curve fitting. Linear regression was used to obtain coefficient vectors that directly map uncorrected data to corrected Lorentzian target parameters. L1 regularization was applied to find subsets of the originally acquired CEST measurements that still allow for such a linear projection mapping. The linear projection method allows fast and interpretable mapping from acquired raw data to contrast parameters of interest, generalizing from healthy subject training data to unseen healthy test data and to the tumor patient dataset. The L1 regularization method shows that a fraction of the acquired CEST measurements is sufficient to preserve tissue contrasts, offering up to 2.8-fold reduction of scan time. Similar observations as for the 7T data can be made for data from a clinical 3T scanner. Being a fast and interpretable computation step, the proposed method is complementary to neural networks, which have been recently employed for similar purposes. The scan time acceleration offered by the L1 regularization (‘CEST-LASSO’) constitutes a step towards better applicability of multi parametric CEST protocols in clinical context.<br
Keywords: chemical exchange saturation transfer (CEST); feature selection; LASSO; APT; NOE; linear projection

Lipschitz dependence of the coefficients on the resolvent and greedy approximation for scalar elliptic problems Dépendance Lipschitz des coefficients comme fonction de la résolvante et approximation greedy pour les problèmes elliptiques scalaires - Choulli, Mourad; Zuazua, Enrique (Comptes Rendus Mathematique, 2016)
We analyze the inverse problem of identifying the diffusivity coefficient of a scalar elliptic equation as a function of the resolvent operator. We prove that, within the class of measurable coefficients, bounded above and below by positive constants, the resolvent determines the diffusivity in an unique manner. Furthermore, we prove that the inverse mapping from resolvent to the coefficient is Lipschitz in suitable topologies. This result plays a key role when applying greedy algorithms to the approximation of parameter-dependent elliptic problems in an uniform and robust manner, independent of the given source terms. In one space dimension, the results can be improved using the explicit expression of solutions, which allows us to link distances between one resolvent and a linear combination of finitely many others and the corresponding distances on coefficients. These results are also extended to multi-dimensional elliptic equations with variable density coefficients. We also point out some possible extensions and open problems.
CRIS-Team Scopus Importer:2021-06-15

Local Elliptic Regularity for the Dirichlet Fractional Laplacian - Biccari, Umberto; Warma, Mahamadi; Zuazua, Enrique (Advanced Nonlinear Studies, 2017)
We prove the Wloc2s,p local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian on an arbitrary bounded open set of RN. The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space. We do it by two different methods. First working directly in the variational formulation of the elliptic problem and then employing the heat kernel representation of solutions.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Dirichlet Boundary Condition; Fractional Laplacian; Local Regularity; Weak Solutions

Localized solutions and filtering mechanisms for the discontinuous Galerkin semi-discretizations of the 1-d wave equation - Marica, Aurora; Zuazua, Enrique (Comptes Rendus Mathematique, 2010)
We perform a complete Fourier analysis of the semi-discrete 1 - d wave equation obtained through a P-1 discontinuous Galerkin (DG) approximation of the continuous wave equation on an uniform grid The resulting system exhibits the interaction of two types of components a physical one and a spurious one related to the possible discontinuities that the numerical solution allows Each dispersion relation contains critical points where the corresponding group velocity vanishes Following previous constructions we rigorously build wave packets with arbitrarily small velocity of propagation concentrated either on the physical or on the spurious component We also develop filtering mechanisms aimed at recovering the uniform velocity of propagation of the continuous solutions Finally some applications to numerical approximation issues of control problems are also presented (C) 2010 Published by Elsevier Masson SAS on behalf of Academie des sciences
CRIS-Team WoS Importer:2019-12-13

Localized solutions for the finite difference semi-discretization of the wave equation - Marica, Aurora; Zuazua, Enrique (Comptes Rendus Mathematique, 2010)
We study the propagation properties of the solutions of the finite difference space semi-discrete wave equation on a uniform grid of the whole Euclidean space. We provide a construction of high frequency wave packets that propagate along the corresponding bi-characteristic rays of Geometric Optics with a group velocity arbitrarily close to zero. Our analysis is motivated by control theoretical issues. In particular, the continuous wave equation has the so-called observability property: for a sufficiently large time, the total energy of its solutions can be estimated in terms of the energy concentrated in the exterior of a compact set. This fails to be true, uniformly on the mesh-size parameter, for the semi-discrete schemes and the observability constant blows-up at an arbitrarily large polynomial order. Our contribution consists in providing a rigorous derivation of those wave packets and in analyzing their behavior near that ray, by taking into account the subtle added dispersive effects that the numerical scheme introduces. (C) 2010 Published by Elsevier Masson SAS on behalf of Academie des sciences.
CRIS-Team WoS Importer:2019-12-13

Local regularity for fractional heat equations - Biccari, Umberto; Warma, Mahamadi; Zuazua, Enrique (SEMA SIMAI Springer Series, 2018)
We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set Ω ⊂ ℝN. Proofs combine classical abstract regularity results for parabolic equations with some new local regularity results for the associated elliptic problems.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Dirichlet boundary condition; Fractional Laplacian; Heat equation; Local regularity; Weak solutions

Local Stability and Convergence of Unconstrained Model Predictive Control - Veldman, Daniel; Zuazua Iriondo, Enrique (2024)
The local stability and convergence for Model Predictive Control (MPC) of unconstrained nonlinear dynamics based on a linear time-invariant plant model is studied. Based on the long-time behavior of the solution of the Riccati Differential Equation (RDE), explicit error estimates are derived that clearly demonstrate the influence of the two critical parameters in MPC: the prediction horizon T and the control horizon τ. In particular, if the MPC-controller has access to an exact (linear) plant model, the MPC-controls and the corresponding optimal state trajectories converge exponentially to the solution of an infinite-horizon optimal control problem when T-τ → ∞. When the difference between the linear model and the nonlinear plant is sufficiently small in a neighborhood of the origin, the MPC strategy is locally stabilizing and the influence of modeling errors can be reduced by choosing the control horizon τ smaller. The obtained convergence rates are validated in numerical simulations.
https://cris.fau.de/converis/publicweb/Publication/276378997
Keywords: Convergence; Model Predictive Control; Receding Horizon Control; Stability

Long-time convergence of a nonlocal Burgers' equation towards the local N-wave - Coclite, Giuseppe Maria; De Nitti, Nicola; Keimer, Alexander; Pflug, Lukas; Zuazua Iriondo, Enrique (Nonlinearity, 2023)
We study the long-time behavior of the unique weak solution of a nonlocal regularization of the (inviscid) Burgers' equation where the velocity is approximated by a one-sided convolution with an exponential kernel. The initial datum is assumed to be positive, bounded, and integrable. The asymptotic profile is given by the "N -wave'' entropy solution of the Burgers' equation. The key ingredients of the proof are a suitable scaling argument and a nonlocal Oleinik-type estimate.
Keywords: Nonlocal conservation laws; nonlocal flux; Burgers’ equation; approximation of local conservation laws; N-waves; source-type solutions; entropy solutions

Long time versus steady state optimal control - Porretta, Alessio; Zuazua, Enrique (SIAM Journal on Control and Optimization, 2013)
This paper analyzes the convergence of optimal control problems for an evolution equation in a finite time-horizon [0, T] toward the limit steady state ones as T ?8. We focus on linear problems. We first consider linear time-independent finite-dimensional systems and show that the optimal controls and states exponentially converge in the transient time (as T tends to infinity) to the ones of the corresponding steady state model. For this to occur suitable observability assumptions need to be imposed. We then extend the results to infinite-dimensional systems including the linear heat and wave equations with distributed controls. Copyright © 2013 Society for Industrial and Applied Mathematics.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Controllability; Long time behavior; Observability; Optimal control problems

Minimal controllability time for finite-dimensional control systems under state constraints - Loheac, Jerome; Trelat, Emmanuel; Zuazua, Enrique (Automatica, 2018)
We consider the controllability problem for finite-dimensional linear autonomous control systems, under state constraints but without imposing any control constraint. It is well known that, under the classical Kalman condition, in the absence of constraints on the state and the control, one can drive the system from any initial state to any final one in an arbitrarily small time. Furthermore, it is also well known that there is a positive minimal time in the presence of compact control constraints. We prove that, surprisingly, a positive minimal time may be required as well under state constraints, even if one does not impose any restriction on the control. This may even occur when the state constraints are unilateral, like the nonnegativity of some components of the state, for instance. Using the Brunovsky normal forms of controllable systems, we analyze this phenomenon in detail, that we illustrate by several examples. We discuss some extensions to nonlinear control systems and formulate some challenging open problems.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Brunovsky normal form; Linear control systems; Minimal time; State constraints

Minimal controllability time for the heat equation under unilateral state or control constraints - Loheac, Jerome; Trelat, Emmanuel; Zuazua, Enrique (Mathematical Models & Methods in Applied Sciences, 2017)
The heat equation with homogeneous Dirichlet boundary conditions is well known to preserve non-negativity. Besides, due to infinite velocity of propagation, the heat equation is null controllable within arbitrary small time, with controls supported in any arbitrarily open subset of the domain (or its boundary) where heat diffuses. The following question then arises naturally: can the heat dynamics be controlled from a positive initial steady state to a positive final one, requiring that the state remains non-negative along the controlled time-dependent trajectory? We show that this state-constrained controllability property can be achieved if the control time is large enough, but that it fails to be true in general if the control time is too short, thus showing the existence of a positive minimal controllability time. In other words, in spite of infinite velocity of propagation, realizing controllability under the unilateral non-negativity state constraint requires a positive minimal time. We establish similar results for unilateral control constraints. We give some explicit bounds on the minimal controllability time, first in 1D by using the sinusoidal spectral expansion of solutions, and then in the multi-dimensional case. We illustrate our results with numerical simulations, and we discuss similar issues for other control problems with various boundary conditions.
CRIS-Team Scopus Importer:2021-06-15
Keywords: constraints; control; Heat process; waiting time

Model predictive control with random batch methods for a guiding problem - Ko, Dongnam; Zuazua Iriondo, Enrique (Mathematical Models & Methods in Applied Sciences, 2021)
We model, simulate and control the guiding problem for a herd of evaders under the action of repulsive drivers. The problem is formulated in an optimal control framework, where the drivers (controls) aim to guide the evaders (states) to a desired region of the Euclidean space. The numerical simulation of such models quickly becomes unfeasible for a large number of interacting agents, as the number of interactions grows O(N2) for N agents. For reducing the computational cost to O(N), we use the Random Batch Method (RBM), which provides a computationally feasible approximation of the dynamics. First, the considered time interval is divided into a number of subintervals. In each subinterval, the RBM randomly divides the set of particles into small subsets (batches), considering only the interactions inside each batch. Due to the averaging effect, the RBM approximation converges to the exact dynamics in the L2-expectation norm as the length of subintervals goes to zero. For this approximated dynamics, the corresponding optimal control can be computed efficiently using a classical gradient descent. The resulting control is not optimal for the original system, but for a reduced RBM model. We therefore adopt a Model Predictive Control (MPC) strategy to handle the error in the dynamics. This leads to a semi-feedback control strategy, where the control is applied only for a short time interval to the original system, and then compute the optimal control for the next time interval with the state of the (controlled) original dynamics. Through numerical experiments we show that the combination of RBM and MPC leads to a significant reduction of the computational cost, preserving the capacity of controlling the overall dynamics.
CRIS-Team Scopus Importer:2021-09-10
Keywords: Agent-based models; guiding problem; large scale complex systems; model predictive control; random batch method

Multilevel control by duality - Biccari, Umberto; Zuazua Iriondo, Enrique (Systems & Control Letters, 2023)
We discuss the multilevel control problem for linear dynamical systems, consisting in designing a piecewise constant control function taking values in a finite set. In particular, we provide a complete characterization of multilevel controls through a duality approach, based on the minimization of a suitable cost functional. In this manner, we devise an effective methodology to systematically build optimal multilevel controls that, depending on the nature of the dynamics considered, have different structural properties. In more detail, in the case of conservative or dissipative dynamics, our technique allows us to define multilevel controls where the levels are fixed a priori, provided that the time horizon is large enough. For more general dynamics satisfying the Kalman rank condition, instead, we are able to build multilevel controls which are effective in any positive time T>0, whose levels blow-up as T→0+. Moreover, this method leads to efficient numerical algorithms for computing multilevel controls.<br
Keywords: Linear control system; Multilevel control; Controllability; Optimal control; Duality

Multilevel Selective Harmonic Modulation by Duality - Biccari, Umberto; Zuazua Iriondo, Enrique (IFAC-PapersOnLine, 2022)
CRIS-Team WoS Importer:2022-10-07

Multilevel selective harmonic modulation by duality - Biccari, Umberto; Zuazua Iriondo, Enrique (2024)
We address the Selective Harmonic Modulation (SHM) problem in power electronic engineering, consisting in designing a multilevel staircase control signal with some prescribed frequencies to improve the performances of a converter. In this work, SHM is addressed through an optimal control methodology based on duality, in which the admissible controls are piece-wise constant functions, taking values only in a given finite set. To fulfill this constraint, the cornerstone of our approach is the introduction of a special penalization in the cost functional, in the form of a piece-wise affine approximation of a parabola. In this manner, we build optimal multilevel controls having the desired staircase structure.<br
https://cris.fau.de/converis/publicweb/Publication/267331774
Keywords: Selective Harmonic Modulation; Optimal Control Theory; Duality; Piece-wise affine penalization; Multilevel controls

Neural ODE Control for Classification, Approximation, and Transport - Ruiz-Balet, Domènec; Zuazua Iriondo, Enrique (SIAM Review, 2023)
We analyze neural ordinary differential equations (NODEs) from a control theoretical perspective to address some of the main properties and paradigms of deep learning (DL), in particular, data classification and universal approximation. These objectives are tackled and achieved from the perspective of the simultaneous control of systems of NODEs. For instance, in the context of classification, each item to be classified corresponds to a different initial datum for the control problem of the NODE, to be classified, all of them by the same common control, to the location (a subdomain of the Euclidean space) associated to each label. Our proofs are genuinely nonlinear and constructive, allowing us to estimate the complexity of the control strategies we develop. The nonlinear nature of the activation functions governing the dynamics of NODEs under consideration plays a key role in our proofs, since it allows deforming half of the phase space while the other half remains invariant, a property that classical models in mechanics do not fulfill. This very property allows us to build elementary controls inducing specific dynamics and transformations whose concatenation, along with properly chosen hyperplanes, allows us to achieve our goals in finitely many steps. The nonlinearity of the dynamics is assumed to be Lipschitz. Therefore, our results apply also in the particular case of the ReLU activation function. We also present the counterparts in the context of the control of neural transport equations, establishing a link between optimal transport and deep neural networks.
CRIS-Team Scopus Importer:2023-09-15
Keywords: data classification; deep learning; neural ODEs; simultaneous control; transport equations; universal approximation; Wasserstein distance

Neural ODE Control for Classification, Approximation and Transport - Ruiz-Balet, Domènec; Zuazua Iriondo, Enrique (2024)
We analyze Neural Ordinary Differential Equations (NODEs) from a control theoretical perspective to address some of the main properties and paradigms of Deep Learning (DL), in particular, data classification and universal approximation. These objectives are tackled and achieved from the perspective of the simultaneous control of systems of NODEs. For instance, in the context of classification, each item to be classified corresponds to a different initial datum for the control problem of the NODE, to be classified, all of them by the same common control, to the location (a subdomain of the euclidean space) associated to each label. Our proofs are genuinely nonlinear and constructive, allowing us to estimate the complexity of the control strategies we develop. The nonlinear nature of the activation functions governing the dynamics of NODEs under consideration plays a key role in our proofs, since it allows deforming half of the phase space while the other half remains invariant, a property that classical models in mechanics do not fulfill. This very property allows to build elementary controls inducing specific dynamics and transformations whose concatenation, along with properly chosen hyperplanes, allows achieving our goals in finitely many steps. The nonlinearity of the dynamics is assumed to be Lipschitz. Therefore, our results apply also in the particular case of the ReLU activation function. We also present the counterparts in the context of the control of neural transport equations, establishing a link between optimal transport and deep neural networks.<br
https://cris.fau.de/converis/publicweb/Publication/259818443
Keywords: Neural ODEs; simultaneous control; universal approximation; data classifi-cation; Deep Learning; transport equations; Wasserstein distance; Optimal Transport.

Nonnegative control of finite-dimensional linear systems - Lohéac, Jérome; Trelat, Emmanuel; Zuazua Iriondo, Enrique (2019)
We consider the controllability problem for nite-dimensional linear autonomous control systems with nonnegative controls. Despite the Kalman condition, the unilateral nonnegativity control constraint may cause a positive minimal controllability time. When this happens, we prove that, if the matrix of the system has a real eigenvalue, then there is a minimal time control in the space of Radon measures, which consists of a nite sum of Dirac impulses. When all eigenvalues are real, this control is unique and the number of impulses is less than half the dimension of the space. We also focus on the control system corresponding to a nite-dierence spatial discretization of the one-dimensional heat equation with Dirichlet boundary controls, and we provide numerical simulations.<br
Keywords: Minimal time; Nonnegative control; Dirac impulse

Nonnegative control of finite-dimensional linear systems - Lohéac, Jérôme; Trélat, Emmanuel; Zuazua Iriondo, Enrique (Annales de l'Institut Henri Poincaré - Analyse Non Linéaire, 2020)
We consider the controllability problem for finite-dimensional linear autonomous control systems with nonnegative controls. Despite the Kalman condition, the unilateral nonnegativity control constraint may cause a positive minimal controllability time. When this happens, we prove that, if the matrix of the system has a real eigenvalue, then there is a minimal time control in the space of Radon measures, which consists of a finite sum of Dirac impulses. When all eigenvalues are real, this control is unique and the number of impulses is less than half the dimension of the space. We also focus on the control system corresponding to a finite-difference spatial discretization of the one-dimensional heat equation with Dirichlet boundary controls, and we provide numerical simulations.
CRIS-Team Scopus Importer:2020-10-09
Keywords: Dirac impulse; Minimal time; Nonnegative control

Norm saturating property of time optimal controls for wave-type equations. This work was partially supported by Grants FA9550-14-1-0214 of the EOARD-AFOSR, FA9550-15-1-0027 of AFOSR and the MTM2014-52347 Grants of the MINECO (Spain) - Loheac, Jerome; Zuazua, Enrique (IFAC-PapersOnLine, 2016)
We consider a time optimal control problem with point target for a class of infinite dimensional systems governed by abstract wave operators. In order to ensure the existence of a time optimal control, we consider controls of energy bounded by a prescribed constant E > 0. Even when this control constraint is absent, in many situations, due to the hyperbolicity of the system under consideration, a target point cannot be reached in arbitrarily small time and there exists a minimal universal controllability time T* > 0, so that for every points y0 and y1 and every time T > T*, there exists a control steering y0 to y1 in time T. Simultaneously this may be impossible if T < T* for some particular choices of y0 and y1. In this note we point out the impact of the strict positivity of the minimal time T* on the structure of the norm of time optimal controls. In other words, the question we address is the following: If T is the minimal time, what is the L2-norm of the associated time optimal control? For different values of y0, y1 and E, we can have τ ≤ T* or τ > T*. If τ > T*, the time optimal control is unique, given by an adjoint problem and its L2-norm is E, in the classical sense. In this case, the time optimal control is also a norm optimal control. But when τ < T*, we show, analyzing the string equation with Dirichlet boundary control, that, surprisingly, there exist time optimal controls which are not of maximal norm E.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Minimal control time; Norm-optimal controls; Open loop control systems; Optimal control; Point-to-point control; Reachable states; Wave equations

Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function - Biccari, Umberto; Zuazua, Enrique (Journal of Differential Equations, 2016)
This article is devoted to the analysis of control properties for a heat equation with a singular potential μ/δ2, defined on a bounded C2 domain Ω⊂RN, where δ is the distance to the boundary function. More precisely, we show that for any μ≤1/4 the system is exactly null controllable using a distributed control located in any open subset of Ω, while for μ>1/4 there is no way of preventing the solutions of the equation from blowing-up. The result is obtained applying a new Carleman estimate.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Carleman estimates; Heat equation; Null controllability; Singular potential

Null controllability for the heat equation with singular inverse-square potentials - Vancostenoble, J.; Zuazua, Enrique (Journal of Functional Analysis, 2008)
We prove the null controllability of the heat equation perturbed by a singular inverse-square potential arising in quantum mechanics and combustion theory. This is done within the range of subcritical coefficients of the singular potential, provided the control acts on an annular set around the singularity. Our proof uses a splitting argument on the domain, decomposition in spherical harmonics, new Carleman inequalities and refined Hardy inequalities. (c) 2008 Elsevier Inc. All rights reserved.
CRIS-Team WoS Importer:2019-12-13

Null controllability for wave equations with memory - Lu, Qi; Zhang, Xu; Zuazua, Enrique (Journal De Mathematiques Pures Et Appliquees, 2017)
We study the memory-type null controllability property for wave equations involving memory terms. The goal is not only to drive the displacement and the velocity (of the considered wave) to rest at some time-instant but also to require the memory term to vanish at the same time, ensuring that the whole process reaches the equilibrium. This memory-type null controllability problem can be reduced to the classical null controllability property for a coupled PDE–ODE system. The latter can be viewed as a degenerate system of wave equations, in which the velocity of propagation for the ODE component vanishes. This fact requires the support of the control to move to ensure the memory-type null controllability to hold, under the so-called Moving Geometric Control Condition. The control result is proved by duality by means of an observability inequality which employs measurements done on a moving observation open subset of the domain where the waves propagate.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Coupled PDE–ODE system; Memory-type null controllability; Moving control; Moving geometric control condition; Wave equations with memory

Null controllability of a system of viscoelasticity with a moving control - Chaves-Silva, Felipe W.; Rosier, Lionel; Zuazua, Enrique (Journal De Mathematiques Pures Et Appliquees, 2014)
In this paper, we consider the wave equation with both viscous Kelvin-Voigt and frictional damping as a model of viscoelasticity in which we incorporate an internal control with a moving support. We prove the null controllability when the control region, driven by the flow of an ODE, covers all the domain. The proof is based upon the interpretation of the system as, roughly, the coupling of a heat equation with an ordinary differential equation (ODE). The presence of the ODE for which there is no propagation along the space variable makes the controllability of the system impossible when the control is confined into a subset in space that does not move. The null controllability of the system with a moving control is established in using the observability of the adjoint system and some Carleman estimates for a coupled system of a parabolic equation and an ODE with the same singular weight, adapted to the geometry of the moving support of the control. This extends to the multi-dimensional case the results by P. Martin et al. in the one-dimensional case, employing 1-d Fourier analysis techniques. © 2013 Elsevier Masson SAS.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Carleman estimates; Decoupling; Moving control; Null controllability; Viscoelasticity

Null controllability of linear heat and wave equations with nonlocal spatial terms - Fernandez-Cara, Enrique; Lu, Qi; Zuazua, Enrique (SIAM Journal on Control and Optimization, 2016)
In this paper, we study the null controllability of linear heat and wave equations with spatial nonlocal integral terms. Under some analyticity assumptions on the corresponding kernel, we show that the equations are controllable. We employ compactness-uniqueness arguments in a suitable functional setting, an argument that is harder to apply for heat equations because of its very strong time irreversibility. Some possible extensions and open problems concerning other nonlocal systems are presented.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Heat equation; Nonlocal terms; Null controllability; Wave equation

Null-controllability of the heat equation in unbounded domains - Micu, Sorin; Zuazua, Enrique (Unsolved Problems in Mathematical Systems and Control Theory, 2009)
CRIS-Team Scopus Importer:2021-06-15

Null controllability of viscous Hamilton-Jacobi equations - Porretta, Alessio; Zuazua, Enrique (Annales de l'Institut Henri Poincaré - Analyse Non Linéaire, 2012)
We study the problem of null controllability for viscous Hamilton-Jacobi equations in bounded domains of the Euclidean space in any space dimension and with controls localized in an arbitrary open nonempty subset of the domain where the equation holds. We prove the null controllability of the system in the sense that, every bounded (and in some cases uniformly continuous) initial datum can be driven to the null state in a sufficiently large time. The proof combines decay properties of the solutions of the uncontrolled system and local null controllability results for small data obtained by means of Carleman inequalities. We also show that there exists a waiting time so that the time of control needs to be large enough, as a function of the norm of the initial data, for the controllability property to hold. We give sharp asymptotic lower and upper bounds on this waiting time both as the size of the data tends to zero and infinity. These results also establish a limit on the growth of nonlinearities that can be controlled uniformly on a time independent of the initial data. (C) 2011 Elsevier Masson SAS. All rights reserved.
CRIS-Team WoS Importer:2019-12-13

NUMERICAL APPROXIMATION OF A ONE-DIMENSIONAL ELLIPTIC OPTIMAL DESIGN PROBLEM - Casado-Diaz, Juan; Castro, Carlos; Luna-Laynez, Manuel; Zuazua, Enrique (Multiscale Modeling & Simulation, 2011)
We address the numerical approximation by finite-element methods of an optimal design problem for a two phase material in one space dimension. This problem, in the continuous setting, due to high frequency oscillations, often does not have a classical solution, and a relaxed formulation is needed to ensure existence. On the contrary, the discrete versions obtained by numerical approximation have a solution. In this article we prove the convergence of the discretizations and obtain convergence rates. We also show a faster convergence when the relaxed version of the continuous problem is taken into account when building the discretization strategy. In particular it is worth emphasizing that, even when the original problem has a classical solution so that relaxation is not necessary, numerical algorithms converge faster when implemented on the relaxed version.
CRIS-Team WoS Importer:2019-12-13

Numerical approximation of null controls for the heat equation : Ill-posedness and remedies - Muench, Arnaud; Zuazua, Enrique (Inverse Problems, 2010)
The numerical approximation of exact or trajectory controls for the wave equation is known to be a delicate issue, since the pioneering work of Glowinski-Lions in the nineties, because of the anomalous behavior of the high-frequency spurious numerical waves. Various efficient remedies have been developed and analyzed in the last decade to filter out these high-frequency components: Fourier filtering, Tychonoff's regularization, mixed finite-element methods, multi-grid strategies, etc. Recently convergence rate results have also been obtained. This work is devoted to analyzing this issue for the heat equation, which is the opposite paradigm because of its strong dissipativity and smoothing properties. The existing analytical results guarantee that, at least in some simple situations, as in the finite-difference scheme in 1 - d, the null or trajectory controls for numerical approximation schemes converge. This is due to the intrinsic high-frequency damping of the heat equation that is inherited by its numerical approximation schemes. But when developing numerical simulations the topic appears to be much more subtle and difficult. In fact, efficiently computing the null control for a numerical approximation scheme of the heat equation is a difficult problem in itself. The difficulty is strongly related to the regularizing effect of the heat kernel. The controls of minimal L(2)-norm are characterized as minima of quadratic functionals on the solutions of the adjoint heat equation, or its numerical versions. These functionals are shown to be coercive in very large spaces of solutions, sufficient to guarantee the L(2) character of controls, but very far from being identifiable as energy spaces for the adjoint system. The very weak coercivity of the functionals under consideration makes the approximation problem exponentially ill-posed and the functional framework far from being well adapted to standard techniques in numerical analysis. In practice, the controls of the minimal L(2)-norm exhibit a singular highly oscillatory behavior near the final controllability time, which cannot be captured numerically. Standard techniques, such as Tychonoff's regularization or quasi-reversibility methods, allow a slight smoothing of the singularities but significantly reduce the quality of the approximation. In this paper we develop some more involved and less-standard approaches which turn out to be more efficient. We first discuss the advantages of using controls with compact support with respect to the time variable or the effect of adding numerical dissipative singular terms. We also develop the numerical version of the so-called transmutation method that allows writing the control of a heat process in terms of the corresponding control of the associated wave process, by means of a 'time convolution' with a one-dimensional controlled fundamental heat solution. This method, although it can be proved to converge, is also subtle in its computational implementation. Indeed, it requires using convergent numerical schemes for the control of the wave equation, a problem that, as mentioned above, is delicate in itself. But we also need to compute an accurate approximation of a controlled fundamental heat solution, an issue that requires its own analysis and significant numerical and computational new developments. These methods are thoroughly illustrated and discussed in the paper, accompanied by some numerical experiments in one space dimension that show the subtlety of the issue.
CRIS-Team WoS Importer:2019-12-13

Numerical approximation schemes for multi-dimensional wave equations in asymmetric spaces - Lescarret, Vincent; Zuazua, Enrique (Mathematics of Computation, 2015)
We develop finite difference numerical schemes for a model arising in multi-body structures, previously analyzed by H. Koch and E. Zuazua, constituted by two n-dimensional wave equations coupled with a (n - 1)- dimensional one along a flexible interface. That model, under suitable assumptions on the speed of propagation in each media, is well-posed in asymmetric spaces in which the regularity of solutions differs by one derivative from one medium to the other. Here we consider a flat interface and analyze this property at a discrete level, for finite difference and mixed finite element methods on regular meshes parallel to the interface. We prove that those methods are well-posed in such asymmetric spaces uniformly with respect to the mesh-size parameters and we prove the convergence of the numerical solutions towards the continuous ones in these spaces. In other words, these numerical methods that are well-behaved in standard energy spaces, preserve the convergence properties in these asymmetric spaces too. These results are illustrated by several numerical experiments.
CRIS-Team Scopus Importer:2021-06-15

Numerical aspects of large-time optimal control of Burgers equation - Allahverdi, Navid; Pozo, Alejandro; Zuazua, Enrique (ESAIM: Mathematical Modelling and Numerical Analysis, 2016)
In this paper, we discuss the efficiency of various numerical methods for the inverse design of the Burgers equation, both in the viscous and in the inviscid case, in long time-horizons. Roughly, the problem consists in, given a final desired target, to identify the initial datum that leads to it along the Burgers dynamics. This constitutes an ill-posed backward problem. We highlight the importance of employing a proper discretization scheme in the numerical approximation of the equation under consideration to obtain an accurate approximation of the optimal control problem. Convergence in the classical sense of numerical analysis does not suffice since numerical schemes can alter the dynamics of the underlying continuous system in long time intervals. As we shall see, this may end up affecting the efficiency on the numerical approximation of the inverse design, that could be polluted by spurious high frequency numerical oscillations. To illustrate this, two well-known numerical schemes are employed: the modified Lax-Friedrichs scheme (MLF) and the Engquist-Osher (EO) one. It is by now well-known that the MLF scheme, as time tends to infinity, leads to asymptotic profiles with an excess of viscosity, while EO captures the correct asymptotic dynamics. We solve the inverse design problem by means of a gradient descent method and show that EO performs robustly, reaching efficiently a good approximation of the minimizer, while MLF shows a very strong sensitivity to the selection of cell and time-step sizes, due to excess of numerical viscosity. The achieved numerical results are confirmed by numerical experiments run with the open source nonlinear optimization package (IPOPT).
CRIS-Team Scopus Importer:2021-06-15
Keywords: Burgers equation; descent method; inverse design; numerics; optimization

Numerical aspects of sonic-boom minimization - Allahverdi, Navid; Pozo, Alejandro; Zuazua, Enrique (Contemporary Mathematics, 2016)
The propagation of the sonic-boom produced by supersonic air-crafts can be modeled by means of a nonlocal version of the viscous Burgers equation. In this paper, motivated by the sonic-boom minimization problem, we analyze the numerical aspects of the optimization process, with a focus on the large-time dynamics of the underlying model. We develop an adjoint methodology at the numerical level allowing to recover accurate approximations of the minimizers. We observe however that some of the minima are hard to achieve due to the intrinsic nonlinear and viscous effects.
CRIS-Team Scopus Importer:2021-06-15

Numerical Control: Part A - Trélat, Emmanuel; Zuazua Iriondo, Enrique (Numerical Control: Part A, 2022)
Numerical Control: Part A, Volume 23 in the Handbook of Numerical Analysis series, highlights new advances in the field, with this new volume presenting interesting chapters written by an international board of authors. Chapters in this volume include Numerics for finite-dimensional control systems, Moments and convex optimization for analysis and control of nonlinear PDEs, The turnpike property in optimal control, Structure-Preserving Numerical Schemes for Hamiltonian Dynamics, Optimal Control of PDEs and FE Approximation, Filtration techniques for the uniform controllability of semi-discrete hyperbolic equations, Numerical controllability properties of fractional partial differential equations, Optimal Control, Numerics, and Applications of Fractional PDEs, and much more.
Keywords: Numerical control; control; nonlinear PDEs; Turnpike Property; optimal control

NUMERICAL DISPERSIVE SCHEMES FOR THE NONLINEAR SCHRODINGER EQUATION - Ignat, Liviu I.; Zuazua, Enrique (SIAM Journal on Numerical Analysis, 2009)
We consider semidiscrete approximation schemes for the linear Schrodinger equation and analyze whether the classical dispersive properties of the continuous model hold for these approximations. For the conservative finite difference semidiscretization scheme we show that, as the mesh size tends to zero, the semidiscrete approximate solutions lose the dispersion property. This fact is proved by constructing solutions concentrated at the points of the spectrum where the second order derivatives of the symbol of the discrete Laplacian vanish. Therefore this phenomenon is due to the presence of numerical spurious high frequencies. To recover the dispersive properties of the solutions at the discrete level, we introduce two numerical remedies: Fourier filtering and a two-grid preconditioner. For each of them we prove Strichartz-like estimates and a local space smoothing effect, uniform in the mesh size. The methods we employ are based on classical estimates for oscillatory integrals. These estimates allow us to treat nonlinear problems with L-2-initial data, without additional regularity hypotheses. We prove the convergence of the two-grid method for nonlinearities that cannot be handled by energy arguments and which, even in the continuous case, require Strichartz estimates.
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Numerical hypocoercivity for the Kolmogorov equation - Porretta, Alessio; Zuazua, Enrique (Mathematics of Computation, 2017)
We prove that a finite-difference centered approximation for the Kolmogorov equation in the whole space preserves the decay properties of continuous solutions as ( → ∞, independently of the mesh-size parameters. This is a manifestation of the property of numerical hypo-coercivity, and it holds both for semi-discrete and fully discrete approximations. The method of proof is based on the energy methods developed by Herau and Villani, employing well-balanced Lyapunov functionals mixing different energies, suitably weighted and equilibrated by multiplicative powers in time. The decreasing character of this Lyapunov functional leads to the optimal decay of the L2- norms of solutions and partial derivatives, which are of different order because of the anisotropy of the model.
CRIS-Team Scopus Importer:2021-06-15

Numerical meshes ensuring uniform observability of one-dimensional waves: Construction and analysis - Ervedoza, Sylvain; Marica, Aurora; Zuazua, Enrique (IMA Journal of Numerical Analysis, 2016)
We build nonuniform numerical meshes for the finite difference and finite element approximations of the one-dimensional wave equation, ensuring that all numerical solutions reach the boundary, as continuous solutions do, in the sense that the full discrete energy can be observed by means of boundary measurements, uniformly with respect to the mesh size. The construction of the nonuniform mesh is achieved by means of a concave diffeomorphic transformation of a uniform grid into a nonuniform one, making the mesh finer and finer when approaching the right boundary. For uniform meshes it is known that high-frequency numerical wave packets propagate very slowly without ever getting to the boundary. Our results show that this pathology can be avoided by taking suitable nonuniform meshes. This also allows us to build convergent numerical algorithms for the approximation of boundary controls of the wave equation.
CRIS-Team Scopus Importer:2021-06-15
Keywords: boundary control; boundary observation; nonuniform meshes; numerical approximation; wave equation

Observability for heat equations with time-dependent analytic memory - Wang, Gengsheng; Zhang, Yubiao; Zuazua Iriondo, Enrique (Archive for Rational Mechanics and Analysis, 2024)
This paper presents a complete analysis of the observability property of heat equations with time-dependent real analytic memory kernels. More precisely, we characterize the geometry of the space-time measurable observation sets ensuring sharp observability inequalities, which are relevant both for control and inverse problems purposes. Despite the abundant literature on the observation of heat-like equations, existing methods do not apply to models involving memory terms. We present a new methodology and observation strategy, relying on the decomposition of the flow, the time-analyticity of solutions and the propagation of singularities. This allows us to obtain a sufficient and necessary geometric condition on the measurable observation sets for sharp two-sided observability inequalities. In addition, some applications to control and relevant open problems are presented.
Keywords: Heat equations with memory; analytic kernels; observability inequalities; flow decomposition; propagation of singularities; geometric characterization of observation sets

Observability of heat processes by transmutation without geometric restrictions - Ervedoza, Sylvain; Zuazua, Enrique (Mathematical Control and Related Fields, 2011)
The goal of this note is to explain how transmutation techniques (originally introduced in [14] in the context of the control of the heat equation, inspired on the classical Kannai transform, and recently revisited in [4] and adapted to deal with observability problems) can be applied to derive observability results for the heat equation without any geometric restriction on the subset in which the control is being applied, from a good understanding of the wave equation. Our arguments are based on the recent results in [15] on the frequency depending observability inequalities for waves without geometric restrictions, an iteration argument recently developed in [13] and the new representation formulas in [4] allowing to make a link between heat and wave trajectories.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Heat equation; Observability; Transmutation

On a nonlocal moving frame approximation of traveling waves - Arrieta, Jose M.; Lopez-Fernandez, Maria; Zuazua, Enrique (Comptes Rendus Mathematique, 2011)
The profiles of traveling wave solutions of a 1-d reaction-diffusion parabolic equation are transformed into equilibria of a nonlocal equation, by means of an appropriate nonlocal change of variables. In this new formulation both the profile and the propagation speed of the traveling waves emerge as asymptotic limits of solutions of a nonlocal reaction-diffusion problem when time goes to infinity. In this Note we make these results rigorous analyzing the well-posedness and the stability properties of the corresponding nonlocal Cauchy problem. We also analyze its restriction to a finite interval with consistent boundary conditions. For large enough intervals we show that there is an asymptotically stable equilibrium which approximates the profile of the traveling wave in R. This leads to efficient numerical algorithms for computing the traveling wave profile and speed of propagation. (C) 2011 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.
CRIS-Team WoS Importer:2019-12-13

On the Controllability of Entropy Solutions of Scalar Conservation Laws at a Junction via Lyapunov Methods - De Nitti, Nicola; Zuazua Iriondo, Enrique (Vietnam Journal of Mathematics, 2023)
In this note, we prove a controllability result for entropy solutions of scalar conservation laws on a star-shaped graph. Using a Lyapunov-type approach, we show that, under a monotonicity assumption on the flux, if u and v are two entropy solutions corresponding to different initial data and same in-flux boundary data (at the exterior nodes of the star-shaped graph), then u ≡ v for a sufficiently large time. In order words, we can drive u to the target profile v in a sufficiently large control time by inputting the trace of v at the exterior nodes as in-flux boundary data for u. This result can also be shown to hold on tree-shaped networks by an inductive argument. We illustrate the result with some numerical simulations.<br
Keywords: Scalar conservation laws; Entropy solutions; Controllability; Lyapunov; Networks; Star-shaped graphs; Tree-shaped graphs

On the decay of one-dimensional locally and partially dissipated and hyperbolic systems - Crin-Barat, Timothee; De Nitti, Nicola; Zuazua Iriondo, Enrique (2025)
We study the time-asymptotic behavior of linear hyperbolic systems under partial dissipation which is localized in suitable subsets of the domain. More precisely, we recover the classical decay rates of partially dissipative systems satisfying the stability condition (SK) with a time-delay depending only on the velocity of each component and the size of the undamped region. To quantify this delay, we assume that the undamped region is a bounded space-interval and that the system without spacerestriction on the dissipation satisfies the stability condition (SK). The former assumption ensures that the time spent by the characteristics of the system in the undamped region is finite and the latter that whenever the damping is active the solutions decay. Our approach consists in reformulating the system into n coupled transport equations and showing that the time-decay estimates are delayed by the sum of the times each characteristics spend in the undamped region.<br
https://cris.fau.de/converis/publicweb/Publication/276246693
Keywords: Hyperbolic systems; partially dissipative hyperbolic systems; localized damping; time-decay; Shizuta-Kawashima condition

ON THE EQUIVALENCE OF MINIMAL TIME AND MINIMAL NORM CONTROLS FOR INTERNALLY CONTROLLED HEAT EQUATIONS - Wang, Gengsheng; Zuazua, Enrique (SIAM Journal on Control and Optimization, 2012)
We prove the equivalence of the minimal time and minimal norm control problems for heat equations on bounded smooth domains of the Euclidean space with homogeneous Dirichlet boundary conditions and controls distributed internally on an open subset of the domain where the equation evolves. We consider the problem of null controllability whose aim is to drive solutions to rest in a finite final time. As a consequence of this equivalence, using the well-known variational characterization of minimal norm controls we establish necessary and sufficient conditions for the minimal time and the corresponding control.
CRIS-Team WoS Importer:2019-12-13

On the lack of controllability of fractional in time ODE and PDE - Lu, Qi; Zuazua, Enrique (Mathematics of Control Signals and Systems, 2016)
This paper is devoted to the analysis of the problem of controllability of fractional (in time) ordinary and partial differential equations (ODE/PDE). The fractional time derivative introduces some memory effects on the system that need to be taken into account when defining the notion of control. In fact, in contrast with the classical ODE and PDE control theory, when driving these systems to rest, one is required not only to control the value of the state at the final time, but also the memory accumulated by the long-tail effects that the fractional derivative introduces. As a consequence, the notion of null controllability to equilibrium needs to take into account both the state and the memory term. The existing literature so far is only concerned with the problem of partial controllability in which the state is controlled, but the behavior of the memory term is ignored. In the present paper, we consider the full controllability problem and show that, due to the memory effects, even at the ODE level, controllability cannot be achieved in finite time. This negative result holds even for finite-dimensional systems in which the control is of full dimension. Consequently, the same negative results hold also for fractional PDE, regardless of their parabolic or hyperbolic nature. This negative result exhibits a completely opposite behavior with respect to the existing literature on classical ODE and PDE control where sharp sufficient conditions for null controllability are well known.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Fractional in time ODE; Null controllability; Observability; Partial controllability; PDE

On the observability of time-discrete conservative linear systems - Ervedoza, Sylvain; Zheng, Chuang; Zuazua, Enrique (Journal of Functional Analysis, 2008)
We consider various time discretization schemes of abstract conservative evolution equations of the form (z) over dot = Az, where A is a skew-adjoint operator. We analyze the problem of observability through an operator B. More precisely, we assume that the pair (A, B) is exactly observable for the continuous model, and we derive uniform observability inequalities for suitable time-discretization schemes within the class of conveniently filtered initial data. The method we use is mainly based on the resolvent estimate given by Burq and Zworski in [N. Burq, M. Zworski, Geometric control in the presence of a black box, J. Amer. Math. Soc. 17(2) (2004) 443-471 (electronic)]. We present some applications of our results to time-discrete schemes for wave, Schrodinger and KdV equations and fully discrete approximation schemes for wave equations. (C) 2008 Elsevier Inc. All rights reserved.
CRIS-Team WoS Importer:2019-12-13

On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials - Duyckaerts, Thomas; Zhang, Xu; Zuazua, Enrique (Annales de l'Institut Henri Poincaré - Analyse Non Linéaire, 2008)
In this paper we prove the optimality of the observability inequality for parabolic systems with potentials in even space dimensions n >= 2. This inequality (derived by E. FernAndez-Cara and the third author in the context of the scalar heat equation with potentials in any space dimension) asserts, roughly, that for small time, the total energy of solutions can be estimated from above in terms of the energy localized in a subdomain with an observability constant of the order of exp(C parallel to a parallel to(2/3)(infinity)), a being the potential involved in the system. The problem of the optimality of the observability inequality remains open for scalar equations.
CRIS-Team WoS Importer:2019-12-13

ON THE QUADRATIC FINITE ELEMENT APPROXIMATION OF ONE-DIMENSIONAL WAVES: PROPAGATION, OBSERVATION, AND CONTROL - Marica, Aurora; Zuazua, Enrique (SIAM Journal on Numerical Analysis, 2012)
We study the propagation, observation, and control properties of the quadratic P-2-classical finite element semidiscretization of the one-dimensional wave equation on a bounded interval. A careful Fourier analysis of the discrete wave dynamics reveals two different branches in the spectrum: the acoustic one, of physical nature, and the optic one, related to the perturbations that this second-order finite element approximation introduces with respect to the P-1 one. On both modes there are high frequencies with vanishing group velocity as the mesh size tends to zero. This shows that the classical property of continuous waves of being observable from the boundary fails to be uniform for this discretization scheme. As a consequence of this, the controls of the discrete waves may blow up as the mesh size tends to zero. To remedy these high-frequency pathologies, we design filtering mechanisms based on the Fourier truncation method or on a bi-grid algorithm, for which one can recover the uniformity of the observability constant in a finite time and, consequently, the possibility to control with uniformly bounded L-2-controls appropriate projections of the solutions. This also allow us to show that, by relaxing the control requirement, the controls are uniformly bounded and converge to the continuous ones as the mesh size tends to zero.
CRIS-Team WoS Importer:2019-12-13

On the regularity of null-controls of the linear 1-d heat equation - Micu, Sorin; Zuazua, Enrique (Comptes Rendus Mathematique, 2011)
The 1-d heat equation in a bounded interval is null-controllable from the boundary. More precisely, for each initial data y(0) is an element of L-2(0.1) there corresponds a unique boundary control of minimal L-2(0. T)-norm which drives the state of the 1-d linear heat equation to zero in time T > 0. This control is given as the normal derivative of a solution of the homogeneous adjoint equation whose initial data (phi) over cap (0) minimizes a suitable quadratic cost functional. In this Note we analyze the relation between the regularity of the initial datum to be controlled y(0) and that of (phi) over cap (0). We show that, if y(0) has only one Fourier mode, the corresponding (phi) over cap (0) does not belong to any Sobolev space of negative exponent. This explains the severe ill-posedness of the problem and the lack of efficiency of most of the existing numerical algorithms for the numerical approximation of the controls, and in particular the slow convergence rate of Tychonoff regularization procedures. (C) 2011 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.
CRIS-Team WoS Importer:2019-12-13

Optimal actuator design via Brunovsky’s normal form - Geshkovski, Borjan; Zuazua Iriondo, Enrique (IEEE Transactions on Automatic Control, 2022)
In this paper, by using the Brunovsky normal form, we provide a reformulation of the problem consisting in finding the actuator design which minimizes the controllability cost for finite-dimensional linear systems with scalar controls. Such systems may be seen as spatially discretized linear partial differential equations with lumped controls. The change of coordinates induced by Brunovsky’s normal form allows us to remove the restriction of having to work with diagonalizable system dynamics, and does not entail a randomization procedure as done in past literature on diffusion equations or waves. Instead, the optimization problem reduces to a minimization of the norm of the inverse of a change of basis matrix, and allows for an easy deduction of existence of solutions, and for a clearer picture of some of the problem’s intrinsic symmetries. Numerical experiments help to visualize these artifacts, indicate further open problems, and also show a possible obstruction of using gradient-based algorithms – this is alleviated by using an evolutionary algorithm.<br
Keywords: Brunovsky normal form; Controllability; finite-dimensional systems; kalman rank condition; lumped control; optimal actuator design

Optimal control and vanishing viscosity for the burgers equation - Castro, Carlos M.; Palacios, F.; Zuazua, Enrique (Integral Methods in Science and Engineering: Analytic Methods, 2010)
CRIS-Team Scopus Importer:2021-06-15

Optimal control approach for moving bottom detection in one-dimensional shallow waters by surface measurements - Lecaros, Rodrigo; Lopez-Rios, J; Montecinos, J; Zuazua Iriondo, Enrique (Mathematical Methods in the Applied Sciences, 2024)
We consider the Boussinesq-Peregrine (BP) system as described by Lannes [Lannes, D. (2013). The water waves problem: mathematical analysis and asymptotics (Vol. 188). American Mathematical Soc.], within the shallow water regime, and study the inverse problem of determining the time and space variations of the channel bottom profile, from measurements of the wave profile and its velocity on the free surface. A well-posedness result within a Sobolev framework for (BP), considering a time dependent bottom, is presented. Then, the inverse problem is reformulated as a nonlinear PDE-constrained optimization one. An existence result of the minimum, under constraints on the admissible set of bottoms, is presented. Moreover, an implementation of the gradient descent approach, via the adjoint method, is considered. For solving numerically both, the forward (BP) and its adjoint system, we derive a universal and low-dissipation scheme, which contains non-conservative products. The scheme is based on the FORCE-α method proposed in [Toro, E. F., Saggiorato, B., Tokareva, S., and Hidalgo, A. (2020). Low-dissipation centred schemes for hyperbolic equations in conservative and non-conservative form. Journal of Computational Physics, 416, 109545]. Finally, we implement this methodology to recover three different bottom profiles; a smooth bottom, a discontinuous one, and a continuous profile with a large gradient. We compare with two classical discretizations for (BP) and the adjoint system. These results corroborate the effectiveness of the proposed methodology to recover bottom profiles.<br

Optimal Control for ODEs and PDEs: The Turnpike Phenomenon - Gugat, Martin; Zuazua Iriondo, Enrique; Schuster, Michael (Extended Abstracts of the 65th Joint Conference on Automatic Control, 2022)
The turnpike phenomenon for dynamic optimal control problems provide insights about the relation between the dynamic optimal control and the solution of the corresponding static optimal control problem. In this talk we give an overview about different turnpike structures for optimal control problems with ordinary differential equations (ODEs) and partial differential equations (PDEs). For optimal control problems with ODEs an exponential turnpike inequality can be shown by basic control theory. These results can be extended to an integral turnpike inequality for optimal control problems with linear hyperbolic systems. For an optimal control problem with non differential tracking term in the objective function, that is exactly controllable, we can show under certain assumptions that the optimal system state is steered exactly to the desired state after finite time. Further we consider an optimal control problem for a hyperbolic system with random boundary data and we show the existence of optimal controls. A turnpike property for hyperbolic systems with random boundary data can be shown numericall
Keywords: Turnpike phenomenon; hyperbolic differential equation; optimal control; boundary control; random boundary data

Optimal convergence rates for the finite element approximation of the Sobolev constant - Ignat, Liviu I.; Zuazua Iriondo, Enrique (2025)
We establish optimal convergence rates for the P1 finite element approximation of the Sobolev constant in arbitrary dimensions N ≥ 2 and for Lebesgue exponents 1 < p < N. Our analysis relies on a refined study of the Sobolev deficit in suitable quasi-norms, which have been introduced and utilized in the context of finite element approximations of the p- Laplacian. The proof further involves sharp estimates for the finite element approximation of Sobolev minimizers
https://cris.fau.de/converis/publicweb/Publication/339346229

Optimal design of sensors via geometric criteria - Ftouhi, Ilias; Zuazua Iriondo, Enrique (Journal of Geometric Analysis, 2023)
We consider a convex set Ω and look for the optimal convex sensor ω ⊂ Ω of a given measure that minimizes the maximal distance to the points of Ω. This problem can be written as follows infdH(ω,Ω) | |ω|= c and ω ⊂ Ω, where c ∈ (0, |Ω|), dH being the Hausdorff distance. We show that the parametrization via the support functions allows us to formulate the geometric optimal shape design problem as an analytic one. By proving a judicious equivalence result, the shape optimization problem is approximated by a simpler minimization of a quadratic function under linear constraints. We then present some numerical results and qualitative properties of the optimal sensors and exhibit an unexpected symmetry breaking phenomeno
Keywords: Shape optimization; Convex geometry; sensor design

Optimal location of controllers for the one-dimensional wave equation - Privat, Yannick; Trelat, Emmanuel; Zuazua, Enrique (Annales de l'Institut Henri Poincaré - Analyse Non Linéaire, 2013)
In this paper, we consider the homogeneous one-dimensional wave equation defined on (0, pi). For every subset omega subset of vertical bar 0, pi vertical bar of positive measure, every T >= 2 pi, and all initial data, there exists a unique control of minimal norm in L-2(0, T L-2(omega)) steering the system exactly to zero. In this article we consider two optimal design problems. Let L is an element of (0, 1). The first problem is to determine the optimal shape and position of omega in order to minimize the norm of the control for given initial data, over all possible measurable subsets omega of vertical bar 0, pi vertical bar of Lebesgue measure L pi. The second problem is to minimize the norm of the control operator, over all such subsets. Considering a relaxed version of these optimal design problems, we show and characterize the emergence of different phenomena for the first problem depending on the choice of the initial data: existence of optimal sets having a finite or an infinite number of connected components, or nonexistence of an optimal set (relaxation phenomenon). The second problem does not admit any optimal solution except for L = 1/2. Moreover, we provide an interpretation of these problems in terms of a classical optimal control problem for an infinite number of controlled ordinary differential equations. This new interpretation permits in turn to study modal approximations of the two problems and leads to new numerical algorithms. Their efficiency will be exhibited by several experiments and simulations. (C) 2013 Elsevier Masson SAS. All rights reserved.
CRIS-Team WoS Importer:2019-12-13

Optimal L^p-approximation of convex sets by convex subsets - Ftouhi, Ilias; Zuazua Iriondo, Enrique; et al. (2025)
Given a convex set Ω of R n, we consider the shape optimization problem of finding a convex subset ω ⊂ Ω, of a given measure, minimizing the p-distance functional Jp(ω) := Z Sn−1 |hΩ − hω| p dHn−1 1 p , where 1 ≤ p < ∞ and hω and hΩ are the support functions of ω and the fixed container Ω, respectively. We prove the existence of solutions and show that this minimization problem Γconverges, when p tends to +∞, towards the problem of finding a convex subset ω ⊂ Ω, of a given measure, minimizing the Hausdorff distance to the convex Ω. In the planar case, we show that the free parts of the boundary of the optimal shapes, i.e., those that are in the interior of Ω, are given by polygonal lines. Still in the 2 − d setting, from a computational perspective, the classical method based on optimizing Fourier coefficients of support functions is not efficient, as it is unable to efficiently capture the presence of segments on the boundary of optimal shapes. We subsequently propose a method combining Fourier analysis and a numerical scheme recently introduced in [4], allowing to obtain accurate results, as demonstrated through numerical experiment
https://cris.fau.de/converis/publicweb/Publication/333431026

Optimal Neumann control for the 1D wave equation: Finite horizon, infinite horizon, boundary tracking terms and the turnpike property - Gugat, Martin; Trelat, Emmanuel; Zuazua, Enrique (Systems & Control Letters, 2016)
We consider a vibrating string that is fixed at one end with Neumann control action at the other end. We investigate the optimal control problem of steering this system from given initial data to rest, in time <span class="formulatext stixSupport mathImg" data-mathurl="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167691116000219&_mathId=si6.gif&_user=111111111&_pii=S0167691116000219&_rdoc=1&_issn=01676911&md5=dc65e1dd969b63a090cdbe8f75af5869" title="Click to view the MathML source">TT, by minimizing an objective functional that is the convex sum of the <span class="formulatext stixSupport mathImg" data-mathurl="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167691116000219&_mathId=si7.gif&_user=111111111&_pii=S0167691116000219&_rdoc=1&_issn=01676911&md5=3079d63e3657e658745b511426583242" title="Click to view the MathML source">L2L2-norm of the control and of a boundary Neumann tracking term. We provide an explicit solution of this optimal control problem, showing that if the weight of the tracking term is positive, then the optimal control action is concentrated at the beginning and at the end of the time interval, and in-between it decays exponentially. We show that the optimal control can actually be written in that case as the sum of an exponentially decaying term and of an exponentially increasing term. This implies that, if the time <span class="formulatext stixSupport mathImg" data-mathurl="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167691116000219&_mathId=si6.gif&_user=111111111&_pii=S0167691116000219&_rdoc=1&_issn=01676911&md5=dc65e1dd969b63a090cdbe8f75af5869" title="Click to view the MathML source">TT is large, then the optimal trajectory approximately consists of three arcs, where the first and the third short-time arcs are transient arcs, and in the middle arc the optimal control and the corresponding state are exponentially close to <span class="formulatext stixSupport mathImg" data-mathurl="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167691116000219&_mathId=si9.gif&_user=111111111&_pii=S0167691116000219&_rdoc=1&_issn=01676911&md5=87ca358085c1b81e4909386689c20041" title="Click to view the MathML source">00. This is an example of a turnpike phenomenon for a problem of optimal boundary control. If <span class="formulatext stixSupport mathImg" data-mathurl="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167691116000219&_mathId=si10.gif&_user=111111111&_pii=S0167691116000219&_rdoc=1&_issn=01676911&md5=688de78ae67c7e8a965a0aab766a0511" title="Click to view the MathML source">T=+∞T=+∞ (infinite time horizon problem), then only the exponentially decaying component of the control remains, and the norms of the optimal control action and of the optimal state decay exponentially in time. In contrast to this situation, if the weight of the tracking term is zero and only the control cost is minimized, then the optimal control is distributed uniformly along the whole interval <span class="formulatext stixSupport mathImg" data-mathurl="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167691116000219&_mathId=si11.gif&_user=111111111&_pii=S0167691116000219&_rdoc=1&_issn=01676911&md5=a670a4f1f8741a261ed1ebcadecadcdf" title="Click to view the MathML source">[0,T][0,T] and coincides with the control given by the Hilbert Uniqueness Method. In addition, we establish a similarity theorem stating that, for every <span class="formulatext stixSupport mathImg" data-mathurl="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167691116000219&_mathId=si12.gif&_user=111111111&_pii=S0167691116000219&_rdoc=1&_issn=01676911&md5=ce5ef695954798ed1cbc42f9aa3209e3" title="Click to view the MathML source">T>0T>0, there exists an appropriate weight <span class="formulatext stixSupport mathImg" data-mathurl="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167691116000219&_mathId=si13.gif&_user=111111111&_pii=S0167691116000219&_rdoc=1&_issn=01676911&md5=416e82ef0c199d9a5bd1e83cb3b8883b" title="Click to view the MathML source">λ<1λ<1 for which the optimal solutions of the corresponding finite horizon optimal control problem and of the infinite horizon optimal control problem coincide along the first part of the time interval <span class="formulatext stixSupport mathImg" data-mathurl="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0167691116000219&_mathId=si14.gif&_user=111111111&_pii=S0167691116000219&_rdoc=1&_issn=01676911&md5=72e2fa577e01e923e71bb0b72e218f4d" title="Click to view the MathML source">[0,2][0,2]. We also discuss the turnpike phenomenon from the perspective of a general framework with a strongly continuous semi-group.
Keywords: Vibrating string; Neumann boundary control; Turnpike phenomenon; Exponential stability; Energy decay; Exact control; Infinite horizon optimal control; Similarity theorem; Receding horizon

Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains - Privat, Yannick; Trelat, Emmanuel; Zuazua, Enrique (Journal of the European Mathematical Society, 2016)
We consider the wave and Schrödinger equations on a bounded open connected subset Ω of a Riemannian manifold, with Dirichlet, Neumann or Robin boundary conditions whenever its boundary is nonempty. We observe the restriction of the solutions to a measurable subset ω of Ω during a time interval [0, T] with T > 0. It is well known that, if the pair (Ω, T) satisfies the Geometric Control Condition (ω being an open set), then an observability inequality holds guaranteeing that the total energy of solutions can be estimated in terms of the energy localized in ω (0, T). We address the problem of the optimal location of the observation subset ω among all possible subsets of a given measure or volume fraction. A priori this problem can be modeled in terms of maximizing the observability constant, but from the practical point of view it appears more relevant to model it in terms of maximizing an average either over random initial data or over large time. This leads us to define a new notion of observability constant, either randomized, or asymptotic in time. In both cases we come up with a spectral functional that can be viewed as a measure of eigenfunction concentration. Roughly speaking, the subset ω has to be chosen so as to maximize the minimal trace of the squares of all eigenfunctions. Considering the convexified formulation of the problem, we prove a no-gap result between the initial problem and its convexified version, under appropriate quantum ergodicity assumptions, and compute the optimal value. Our results reveal intimate relations between shape and domain optimization, and the theory of quantum chaos (more precisely, quantum ergodicity properties of the domain Ω). We prove that in 1D a classical optimal set exists only for exceptional values of the volume fraction, and in general one expects relaxation to occur and therefore classical optimal sets not to exist. We then provide spectral approximations and present some numerical simulations that fully confirm the theoretical results in the paper and support our conjectures. Finally, we provide several remedies to nonexistence of an optimal domain.We prove that when the spectral criterion is modified to consider a weighted one in which the high frequency components are penalized, the problem has a unique classical solution determined by a finite number of low frequency modes. In particular the maximizing sequence built from spectral approximations is stationary.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Ergodic properties; Observability inequality; Optimal design; Quantum ergodicity; Schrödinger equation; Spectral decomposition; Wave equation

Optimal Observation of the One-dimensional Wave Equation - Privat, Yannick; Trelat, Emmanuel; Zuazua, Enrique (Journal of Fourier Analysis and Applications, 2013)
In this paper, we consider the homogeneous one-dimensional wave equation on [0,pi] with Dirichlet boundary conditions, and observe its solutions on a subset omega of [0,pi]. Let La(0,1). We investigate the problem of maximizing the observability constant, or its asymptotic average in time, over all possible subsets omega of [0,pi] of Lebesgue measure L pi. We solve this problem by means of Fourier series considerations, give the precise optimal value and prove that there does not exist any optimal set except for L=1/2. When L not equal 1/2 we prove the existence of solutions of a relaxed minimization problem, proving a no gap result. Following H,brard and Henrot (Syst. Control Lett., 48:199-209, 2003; SIAM J. Control Optim., 44:349-366, 2005), we then provide and solve a modal approximation of this problem, show the oscillatory character of the optimal sets, the so called spillover phenomenon, which explains the lack of existence of classical solutions for the original problem.
CRIS-Team WoS Importer:2019-12-13

Optimal Placement and Shape Design of Sensors via Geometric Criteria - Ftouhi, Ilias; Zuazua Iriondo, Enrique (2024 European Control Conference, ECC 2024, 2024)
The optimal placement and design of sensors is commonly encountered in industrial and applied problems, such as urban planning and the supervision of temperature and pressure in gas networks. In essence, sensors are considered optimally designed when they ensure the highest level of observation for the specific phenomenon in question. Typically, this design process is guided by specific objectives and is subject to constraints commonly defined by an appropriate partial differential equation (PDE), taking into account the underlying physics of the process. In the present work, we focus on two independent study cases: The optimal shape design of a convex sensor. • The optimal placement of a finite number of sensors inside a given region. Here, we address the problem in a purely geometric setting, without involving a specific PDE model. We consider a simple and natural geometric criterion of performance, based on distance functions. But, as we shall see, tackling it will require to employ geometric analysis methods.
CRIS-Team Scopus Importer:2024-08-16
Keywords: Temperature sensors; Shape; Partial differential equations; Urban planning; Europe; Sensor phenomena and characterization; Physics

Optimal Shape and Location of Sensors for Parabolic Equations with Random Initial Data - Privat, Yannick; Trelat, Emmanuel; Zuazua, Enrique (Archive for Rational Mechanics and Analysis, 2015)
In this article, we consider parabolic equations on a bounded open connected subset Rn. We model and investigate the problem of optimal shape and location of the observation domain having a prescribed measure. This problem is motivated by the question of knowing how to shape and place sensors in some domain in order to maximize the quality of the observation: for instance, what is the optimal location and shape of a thermometer? We show that it is relevant to consider a spectral optimal design problem corresponding to an average of the classical observability inequality over random initial data, where the unknown ranges over the set of all possible measurable subsets of ΩΩ of fixed measure. We prove that, under appropriate sufficient spectral assumptions, this optimal design problem has a unique solution, depending only on a finite number of modes, and that the optimal domain is semi-analytic and thus has a finite number of connected components. This result is in strong contrast with hyperbolic conservative equations (wave and Schrödinger) studied in Privat et al. (J Eur Math Soc, 2015) for which relaxation does occur. We also provide examples of applications to anomalous diffusion or to the Stokes equations. In the case where the underlying operator is any positive (possible fractional) power of the negative of the Dirichlet-Laplacian, we show that, surprisingly enough, the complexity of the optimal domain may strongly depend on both the geometry of the domain and on the positive power. The results are illustrated with several numerical simulations.
CRIS-Team Scopus Importer:2021-06-15

Optimal shape and location of sensors or actuators in PDE models - Privat, Yannick; Trelat, Emmanuel; Zuazua, Enrique (Proceedings of the American Control Conference, 2014)
We investigate the problem of optimizing the shape and location of sensors and actuators for evolution systems driven by distributed parameter systems or partial differential equations (PDE). We consider wave, Schrödinger and heat equations on an arbitrary domain Ω, in any space dimension, and with suitable boundary conditions (if there is a boundary) which can be of Dirichlet, Neumann, mixed or Robin type. This kind of problem is frequently encountered in applications where one aims, for instance, at maximizing the quality of reconstruction of the solution, using only a partial observation. From the mathematical point of view, using probabilistic considerations we model this problem as that of maximizing the so-called randomized observability constant, over all possible subdomains of Ω having a prescribed measure. The spectral analysis of this problem reveals intimate connections with the theory of quantum chaos. More precisely, we provide a solution to this problem when the domain Ω satisfies suitable quantum ergodicity assumptions. © 2014 American Automatic Control Council.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Distributed parameter systems; Observers for linear systems; Optimization

Optimal strategies for driving a mobile agent in a "guidance by repulsion" model - Escobedo, Ramon; Ibanez, A.; Zuazua, Enrique (Communications in Nonlinear Science and Numerical Simulation, 2016)
We present a guidance by repulsion model based on a driver-evader interaction where the driver, assumed to be faster than the evader, follows the evader but cannot be arbitrarily close to it, and the evader tries to move away from the driver beyond a short distance. The key ingredient allowing the driver to guide the evader is that the driver is able to display a circumvention maneuver around the evader, in such a way that the trajectory of the evader is modified in the direction of the repulsion that the driver exerts on the evader. The evader can thus be driven towards any given target or along a sufficiently smooth path by controlling a single discrete parameter acting on driver's behavior. The control parameter serves both to activate/deactivate the circumvention mode and to select the clockwise/counterclockwise direction of the circumvention maneuver. Assuming that the circumvention mode is more expensive than the pursuit mode, and that the activation of the circumvention mode has a high cost, we formulate an optimal control problem for the optimal strategy to drive the evader to a given target. By means of numerical shooting methods, we find the optimal open-loop control which reduces the number of activations of the circumvention mode to one and which minimizes the time spent in the active mode. Our numerical simulations show that the system is highly sensitive to small variations of the control function, and that the cost function has a nonlinear regime which contributes to the complexity of the behavior of the system, so that a general open-loop control would not be of practical interest. We then propose a feedback control law that corrects from deviations while preventing from an excessive use of the circumvention mode, finding numerically that the feedback law significantly reduces the cost obtained with the open-loop control.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Driver-evader agents; Feedback control law; Guidance by repulsion; Nonlinear dynamics; Numerical simulations; Optimal strategies

Optimized classification with neural ODEs via separability - Alvarez-Lopez, Antonio; Rafael, Orive-Illera; Zuazua Iriondo, Enrique (2024)
Classification of N points becomes a simultaneous control problem when viewed through the lens of neural ordinary differential equations (neural ODEs), which represent the time-continuous limit of residual networks. For the narrow model, with one neuron per hidden layer, it has been shown that the task can be achieved using O(N) neurons. In this study, we focus on estimating the number of neurons required for efficient clusterbased classification, particularly in the worst-case scenario where points are independently and uniformly distributed in [0, 1]d . Our analysis provides a novel method for quantifying the probability of requiring fewer than O(N) neurons, emphasizing the asymptotic behavior as both d and N increase. Additionally, under the sole assumption that the data are in general position, we propose a new constructive algorithm that simultaneously classifies clusters of d points from any initial configuration, effectively reducing the maximal complexity to O(N/d) neurons.<br
https://cris.fau.de/converis/publicweb/Publication/317342024
Keywords: Classification; neural ODEs; simultaneous control; complexity; separability; worst-case scenario

Pattern control via diffusion interaction - Ruiz-Balet, Domènec; Zuazua Iriondo, Enrique (2024)
We analyse a dynamic control problem for scalar reaction-diffusion equations, focusing on the emulation of pattern formation through the selection of appropriate active controls. While boundary controls alone prove inadequate for replicating the complex patterns seen in biological systems, particu- larly under natural point-wise constraints of the system state, their combination with the regulation of the diffusion coefficient enables the successful generation of such patterns. Our study demonstrates that the set of steady-states is path-connected, facilitating the use of the staircase method. This approach allows any admissible initial configuration to evolve into any stationary pattern over a sufficiently long time while maintaining the system’s natural bilateral constraints. We provide also examples of complex patterns that steady-state configurations can adopt.
https://cris.fau.de/converis/publicweb/Publication/325045502

Perfectly matched layers in 1-d : energy decay for continuous and semi-discrete waves - Ervedoza, S.; Zuazua, Enrique (Numerische Mathematik, 2008)
In this paper we investigate the efficiency of the method of perfectly matched layers (PML) for the 1-d wave equation. The PML method furnishes a way to compute solutions of the wave equation for exterior problems in a finite computational domain by adding a damping term on the matched layer. In view of the properties of solutions in the whole free space, one expects the energy of solutions obtained by the PML method to tend to zero as t -> infinity, and the rate of decay can be understood as a measure of the efficiency of the method. We prove, indeed, that the exponential decay holds and characterize the exponential decay rate in terms of the parameters and damping potentials entering in the implementation of the PML method. We also consider a space semi-discrete numerical approximation scheme and we prove that, due to the high frequency spurious numerical solutions, the decay rate fails to be uniform as the mesh size parameter h tends to zero. We show however that adding a numerical viscosity term allows us to recover the property of exponential decay of the energy uniformly on h. Although our analysis is restricted to finite differences in 1-d, most of the methods and results apply to finite elements on regular meshes and to multi-dimensional problems.
CRIS-Team WoS Importer:2019-12-13

Phase portrait control for 1D monostable and bistable reaction-diffusion equations - Pouchol, Camille; Trelat, Emmanuel; Zuazua, Enrique (Nonlinearity, 2019)
We consider the problem of controlling parabolic semilinear equations arising in population dynamics, either in finite time or infinite time. These are the monostable and bistable equations on (0, L) for a density of individuals 0 ≤ y(t, x) ≤ 1, with Dirichlet controls taking their values in [0, 1]. We prove that the system can never be steered to extinction (steady state 0) or invasion (steady state 1) in finite time, but is asymptotically controllable to 1 independently of the size L, and to 0 if the length L of the interval domain is less than some threshold value L, which can be computed from transcendental integrals. In the bistable case, controlling to the other homogeneous steady state 0 < θ< 1 is much more intricate. We rely on a staircase control strategy to prove that can be reached in finite time if and only if L < L. The phase plane analysis of those equations is instrumental in the whole process. It allows us to read obstacles to controllability, compute the threshold value for domain size as well as design the path of steady states for the control strategy.
CRIS-Team Scopus Importer:2021-06-15
Keywords: control; constraints; phase plane; reaction-diffusion; steady state

Preface: A tribute to professor Eduardo Casas on his 60th birthday - Alberto Fernandez, Luis; Mateos, Mariano; Pola, Cecilia; Troeltzsch, Fredi; Zuazua, Enrique (Mathematical Control and Related Fields, 2018)
CRIS-Team Scopus Importer:2021-06-15

Progress and future directions in machine learning through control theory - Zuazua Iriondo, Enrique (Proceedings FGS 20204, 2024)
This paper presents our recent advancements at the intersection of machine learning and control theory. We focus specifically on utilizing control theoretical tools to elucidate the underlying mechanisms driving the success of machine learning algorithms. By enhancing the explainability of these algorithms, we aim to contribute to their ongoing improvement and more effective application. Our research explores several critical areas: Firstly, we investigate the memorization, representation, classification, and approximation properties of residual neural networks (ResNets). By framing these tasks as simultaneous or ensemble control problems, we have developed nonlinear and constructive algorithms for training. Our work provides insights into the parameter complexity and computational requirements of ResNets. Similarly, we delve into the properties of neural ODEs (NODEs). We demonstrate that autonomous NODEs of sufficient width can ensure approximate memorization properties. Furthermore, we prove that by allowing biases to be time-dependent, NODEs can track dynamic data. This showcases their potential for synthetic model generation and helps elucidate the success of methodologies such as Reservoir Computing. Next, we analyze the optimal architectures of multilayer perceptrons (MLPs). Our findings offer guidelines for designing MLPs with minimal complexity, ensuring efficiency and effectiveness for supervised learning tasks. The generalization and prediction capacity of trained networks plays a crucial role. To address these properties, we present two nonconvex optimization problems related to shallow neural networks, capturing the ”sparsity” of parameters and robustness of representation. We introduce a ”mean-field” model, proving, via representer theorems, the absence of a relaxation gap. This aids in designing an optimal tolerance strategy for robustness and, through convexification, efficient algorithms for training. In the context of large language models (LLMs), we explore the integration of residual networks with self-attention layers for context capture. We treat ”attention” as a dynamical system acting on a collection of points and characterize their asymptotic dynamics, identifying convergence towards special points called leaders. These theoretical insights have led to the development of an interpretable model for sentiment analysis of movie reviews, among other possible applications. Lastly, we address federated learning, which enables multiple clients to collaboratively train models without sharing private data, thus addressing data collection and privacy challenges. We examine training efficiency, incentive mechanisms, and privacy concerns within this framework, proposing solutions to enhance the effectiveness and security of federated learning methods. Our work underscores the potential of applying control theory principles to improve machine learning models, resulting in more interpretable and efficient algorithms. This interdisciplinary approach opens up a fertile ground for future research, raising profound mathematical questions and application-oriented challenges and opportunities.<br

Propagation of 1D Waves in Regular Discrete Heterogeneous Media: A Wigner Measure Approach - Marica, Aurora; Zuazua, Enrique (Foundations of Computational Mathematics, 2015)
In this article, we describe the propagation properties of the one-dimensional wave and transport equations with variable coefficients semi-discretized in space by finite difference schemes on non-uniform meshes obtained as diffeomorphic transformations of uniform ones. In particular, we introduce and give a rigorous meaning to notions like the principal symbol of the discrete wave operator and the corresponding bi-characteristic rays. The main mathematical tool we employ is the discrete Wigner transform, which, in the limit as the mesh size parameter tends to zero, yields the so-called Wigner (semiclassical) measure. This measure provides the dynamics of the bi-characteristic rays, i.e., the solutions of the Hamiltonian system describing the propagation, in both physical and Fourier spaces, of the energy of the solution to the wave equation. We show that, due to dispersion phenomena, the high-frequency numerical dynamics does not coincide with the continuous one. Our analysis holds for the class C^0,1(R)C0,1(R) of globally Lipschitz coefficients and non-uniform grids obtained by means of C^1,1(R)C1,1(R)-diffeomorphic transformations of a uniform one. We also present several numerical simulations that confirm the predicted paths of the space–time projections of the bi-characteristic rays. Based on the theoretical analysis and simulations, we describe some of the pathological phenomena that these rays might exhibit as, for example, their reflection before touching the boundary of the space domain. This leads, in particular, to the failure of the classical properties of boundary observability of continuous waves, arising in control and inverse problems theory.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Bi-characteristic rays; Finite difference approximations; Non-uniform mesh; Umklapp/internal reflection; Variable coefficients transport/wave equations; Wigner transform/measure

Propagation of One- and Two-Dimensional Discrete Waves Under Finite Difference Approximation - Biccari, Umberto; Marica, Aurora; Zuazua Iriondo, Enrique (Foundations of Computational Mathematics, 2020)
We analyze the propagation properties of the numerical versions of one- and two-dimensional wave equations, semi-discretized in space by finite difference schemes. We focus on high-frequency solutions whose propagation can be described, both at the continuous and at the semi-discrete levels, by micro-local tools. We consider uniform and non-uniform numerical grids as well as constant and variable coefficients. The energy of continuous and semi-discrete high-frequency solutions propagates along bi-characteristic rays, but their dynamics are different in the continuous and the semi-discrete setting, because of the nature of the corresponding Hamiltonians. One of the main objectives of this paper is to illustrate through accurate numerical simulations that, in agreement with micro-local theory, numerical high-frequency solutions can bend in an unexpected manner, as a result of the accumulation of the local effects introduced by the heterogeneity of the numerical grid. These effects are enhanced in the multi-dimensional case where the interaction and combination of such behaviors in the various space directions may produce, for instance, the rodeo effect, i.e., waves that are trapped by the numerical grid in closed loops, without ever getting to the exterior boundary. Our analysis allows to explain all such pathological behaviors. Moreover, the discussion in this paper also contributes to the existing theory about the necessity of filtering high-frequency numerical components when dealing with control and inversion problems for waves, which is based very much on the theory of rays and, in particular, on the fact that they can be observed when reaching the exterior boundary of the domain, a key property that can be lost through numerical discretization.
CRIS-Team Scopus Importer:2020-02-04
Keywords: Finite difference approximation; Propagation of solutions; Uniform and non-uniform meshes; Wave equation

Randomised observation, control and stabilization of waves: [Based on the plenary lecture presented at the 86th Annual GAMM Conference, Lecce, Italy, March 24, 2015] - Privat, Yannick; Trelat, Emmanuel; Zuazua, Enrique (ZAMM - Zeitschrift für angewandte Mathematik und Mechanik, 2016)
The problems of observing, controlling and stabilizing wave processes arise in many different contexts ranging from structural mechanics to seismic waves. In a suitable functional setting, they are closely interconnected and sometimes completely equivalent. In a series of previous articles we have addressed the problem of the optimal design of sensors for purely conservative wave models. We analyzed a relaxed version of the optimal observation problem, considering the expectation of solutions under a randomisation procedure, rather than that where all possible solutions are considered in a purely deterministic setting. From an analytical point of view, this randomisation procedure had the advantage of leading to a spectral diagonalisation of the observations. In this way, using fine asymptotic spectral properties of the Laplacian, we disclosed the links between the geometric properties of the domain where waves propagate and the existence of optimal locations for the sensors or, by the contrary, the emergence of relaxation phenomena. Here we show that spectral randomised observability is equivalent to the property of spectral controllability by means of a discrete set of lumped controls acting everywhere on the domain, and distributed according to the shape of the eigenfunctions. Our results on optimal observation then find natural equivalents on the problem of optimal spectral control. We also give an interpretation of these results in terms of a feedback stabilization property, ensuring the exponential decay of the energy of solutions as time tends to infinity. The problems of observing, controlling and stabilizing wave processes arise in many different contexts ranging from structural mechanics to seismic waves. In a suitable functional setting, they are closely interconnected and sometimes completely equivalent. In a series of previous articles The authors have addressed the problem of the optimal design of sensors for purely conservative wave models. They analyzed a relaxed version of the optimal observation problem, considering the expectation of solutions under a randomisation procedure, rather than that where all possible solutions are considered in a purely deterministic setting. From an analytical point of view, this randomisation procedure had the advantage of leading to a spectral diagonalisation of the observations.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Abstract conservative semigroup; Exponential decay; Observability; Optimal design; Randomisation; Spectral control; Spectral decomposition; Stabilization; Wave equation

Reachable set for Hamilton-Jacobi equations with non-smooth Hamiltonian and scalar conservation laws - Esteve-Yague, Carlos; Zuazua Iriondo, Enrique (Nonlinear Analysis: Hybrid Systems, 2023)
We give a full characterization of the range of the operator which associates, to any initial condition, the viscosity solution at time T of a Hamilton-Jacobi equation with convex Hamiltonian. Our main motivation is to be able to treat the case of convex Hamiltonians with no further regularity assumptions. We give special attention to the case H(p)=|p|, for which we provide a rather geometrical description of the range of the viscosity operator by means of an interior ball condition on the sublevel sets. From our characterization of the reachable set, we are able to deduce further results concerning, for instance, sharp regularity estimates for the reachable functions, as well as structural properties of the reachable set. The results are finally adapted to the case of scalar conservation laws in dimension one.<br
Keywords: Hamilton-Jacobi equation; inverse design problem; reachable set

Reachable subspaces, control regions and heat equations with memory - Wang, Gengshen; Zhang, Yubiao; Zuazua Iriondo, Enrique (2024)
We study the controlled heat equations with analytic memory from two perspectives:reachable subspaces and control regions. Due to the hybrid parabolic-hyperbolic phenomenonof the equations, the support of a control needs to move in time to efficiently control thedynamics. We show that under a sharp sufficient geometric condition imposed to the controlregions, the difference between reachable subspaces of the controlled heat equations, withand without memory, is precisely given by a Sobolev space. The appearance of this Sobolevspace is attributed to the memory which makes the equation having the wave-like nature.The main ingredients in the proofs of our main results are as: first, the decomposition of theflow (generated by the equation with the null control) given in [31], second, an observabilityinequality built up in this pape
https://cris.fau.de/converis/publicweb/Publication/259277004
Keywords: Controlled heat equations with memory; reachable subspaces; control regions

Recovery of an initial temperature from discrete sampling - Devore, Ronald; Zuazua, Enrique (Mathematical Models & Methods in Applied Sciences, 2014)
The problem of recovering the initial temperature of a body from discrete temperature measurements made at later times is studied. While this problem has a general formulation, the results of this paper are only given in the simplest setting of a finite (one-dimensional), constant coefficient, linear rod. It is shown that with a judicious placement of a thermometer on this rod, the initial temperature profile of the rod can be completely determined by later time measurements. The paper then studies the number of measurements that are needed to recover the initial profile to a prescribed accuracy and provides an optimal reconstruction algorithm under the assumption that the initial profile is in a Sobolev class. © 2014 World Scientific Publishing Company.
CRIS-Team Scopus Importer:2021-06-15
Keywords: discrete time sampling; Fourier representation; Heat equation; initial datum; inversion; optimal recovery; sensing

Regional and Partial Observability and Control of Waves - Dehman, Belhassen; Ervedoza, Sylvain; Zuazua Iriondo, Enrique (2025)
We establish sharp regional observability results for solutions of the wave equation in a bounded domain Ω ⊂ R n , in arbitrary spatial dimension. Assuming the waves are observed on a non-empty open subset ω ⊂ Ω and that the initial data are supported in another open subset O ⊂ Ω, we derive estimates for the energy of initial data localized in O, in terms of the energy measured on the observation set (0, T) × ω. This holds under a suitable geometric condition relating the time horizon T and the pair of subdomains (ω, O). Roughly speaking, this geometric condition requires that all rays of geometric optics in Ω, emanating from O, must reach the observation region (0, T) × ω. Our result significantly generalizes classical observability results, which recover the total energy of all solutions when the observation set ω satisfies the so-called Geometric Control Condition (GCC)—a particular case corresponding to O = Ω. A notable feature of our approach is that it remains effective in settings where Holmgren’s uniqueness does not guarantee unique continuation. As a consequence of our analysis, unique continuation is nonetheless recovered for wave solutions observed on (0, T) × ω with initial data supported in O. The proof of this previously unnoticed result combines a high-frequency observability estimate-based on the propagation of singularities—with a compactness-uniqueness argument that exploits the unique continuation properties of elliptic operators. By duality, this observability result leads to new controllability results for the wave equation, ensuring that the projection of the solution onto O can be controlled by means of controls supported in ω, with optimal spatial support. We also present several extensions of the main result, including the case of boundary observations, as well as a characterization of the observable fraction of the energy of the initial data from partial measurements on (0, T) × ω. Applications to wave control are discussed accordingl
https://cris.fau.de/converis/publicweb/Publication/340593559

Regularity issues for the null-controllability of the linear 1-d heat equation - Micu, Sorin; Zuazua, Enrique (Systems & Control Letters, 2011)
The fact that the heat equation is controllable to zero in any bounded domain of the Euclidean space, any time T > 0 and from any open subset of the boundary is well known. On the other hand, numerical experiments show the ill-posedness of the problem. In this paper we develop a rigorous analysis of the 1-d problem which provides a sharp description of this ill-posedness.
CRIS-Team WoS Importer:2019-12-13

Remarks on long time versus steady state optimal control - Porretta, Alessio; Zuazua, Enrique (Springer International Publishing, 2016)
Control problems play a key role in many fields of Engineering, Economics and Sciences. This applies, in particular, to climate sciences where, often times, relevant problems are formulated in long time scales. The problem of the possible asymptotic simplification (as time tends to infinity) then emerges naturally. More precisely, assuming, for instance, that the free dynamics under consideration stabilizes towards a steady state solution, the following question arises: Do time averages of optimal controls and trajectories converge to the steady optimal controls and states as the time-horizon tends to infinity? This question is very closely related to the so-called turnpike property stating that, often times, the optimal trajectory joining two points that are far apart, consists in, departing from the point of origin, rapidly getting close to the steady-state (the turnpike) to stay there most of the time, to quit it only very close to the final destination and time. In this paper we focus on the semilinear heat equation. We prove some partial results and enumerate a number of interesting topics of future research, indicating also some connections with shape design and inverse problems theory.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Controllability; Long time behavior; Observability; Optimal control problems; Semilinear heat equations; Steady states; Turnpike property

Representation and Regression Problems in Neural Networks: Relaxation, Generalization, and Numerics - Liu, Kang; Zuazua Iriondo, Enrique (Mathematical Models & Methods in Applied Sciences, 2025)
In this work, we address three non-convex optimization problems associated with the training of shallow neural networks (NNs) for exact and approximate representation, as well as for regression tasks. Through a mean-field approach, we convexify these problems and, applying a representer theorem, prove the absence of relaxation gaps. We establish generalization bounds for the resulting NN solutions, assessing their predictive performance on test datasets, analyzing the impact of key hyperparameters on these bounds, and proposing optimal choices. On the computational side, we examine the discretization of the convexified problems and derive convergence rates. For low-dimensional datasets, these discretized problems are efficiently solvable using the simplex method. For high-dimensional datasets, we propose a sparsification algorithm that, combined with gradient descent for over-parameterized shallow NNs, yields effective solutions to the primal problem
CRIS-Team Scopus Importer:2025-04-11
Keywords: generalization; mean-field relaxation; Neural network; numerical algorithm; representer theorem

Robust Grid Adaptation for Efficient Uncertainty Quantification - Palacios, Francisco; Duraisamy, Karthik; Alonso, Juan J.; Zuazua, Enrique (Aiaa Journal, 2012)
In the recent past, adjoint methods have been successfully applied in error estimation of integral outputs (functionals) of the numerical solution of partial differential equations. The adjoint solution can also be used as a grid adaptation indicator, with the objective of optimally targeting and reducing the numerical error in the functional of interest below a prespecified threshold. In situations where we seek to quantify the effect of aleatory uncertainties on statistical moments of the output functional, it becomes necessary to evaluate the functional accurately at multiple sample points in probability space. If the numerical accuracy of these sample evaluations is not uniform, variations in the numerical error can affect the evaluation of the statistical moments. Although it is possible to independently adapt the meshes to obtain more accurate solutions at each sample point in stochastic space, such a procedure can be both cumbersome and computationally expensive. To improve the efficiency of this process, a new robust grid adaptation technique is proposed that is aimed at minimizing the numerical error over a range of variations of the uncertain parameters of interest about a nominal state. Using this approach, it is possible to generate computational grids that are insensitive to small variations of the uncertain parameters that can both locally and globally change the solution and, as a result, the error distribution. This is in contrast with classical adjoint techniques, which seek to adapt the grid with the aim of minimizing numerical errors for a specific flow condition (and geometry). It is demonstrated that flow computations on these robust grids result in low numerical errors under the expected range of variations of the uncertain input parameters. The effectiveness of this strategy is demonstrated in problems involving the Poisson equation and the Euler equations at transonic and supersonic/hypersonic speeds.
CRIS-Team WoS Importer:2019-12-13

Robust null controllability for heat equations with unknown switching control mode - Lu, Qi; Zuazua, Enrique (Discrete and Continuous Dynamical Systems, 2014)
We analyze the null controllability for heat equations in the presence of switching controls. The switching pattern is a priori unknown so that the control has to be designed in a robust manner, based only on the past dynamics, so to fulfill the final control requirement, regardless of what the future dynamics is. We prove that such a robust control strategy actually exists when the switching controllers are located on two non trivial open subsets of the domain where the heat process evolves. Our strategy to construct these robust controls is based on earlier works by Lebeau and Robbiano on the null controllability of the heat equation. It is relevant to emphasize that our result is specific to the heat equation as an extension of a property of finite-dimensional systems that we fully characterize but that it may not hold for wave-like equations.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Heat equations; Null controllability; Robustness; Switching control

Sensitivity analysis of 1-d steady forced scalar conservation laws - Ersoy, Mehmet; Feireisl, Eduard; Zuazua, Enrique (Journal of Differential Equations, 2013)
We analyze 1 - d forced steady state scalar conservation laws. We first show the existence and uniqueness of entropy solutions as limits as t -> infinity of the corresponding solutions of the scalar evolutionary hyperbolic conservation law. We then linearize the steady state equation with respect to perturbations of the forcing term. This leads to a linear first order differential equation with, possibly, discontinuous coefficients. We show the existence and uniqueness of solutions in the context of duality solutions. We also show that this system corresponds to the steady state version of the linearized evolutionary hyperbolic conservation law. This analysis leads us to the study of the sensitivity of the shock location with respect to variations of the forcing term, an issue that is relevant in applications to optimal control and parameter identification problems. (C) 2013 Elsevier Inc. All rights reserved.
CRIS-Team WoS Importer:2019-12-13

Shape turnpike for linear parabolic PDE models - Lance, Gontran; Trélat, Emmanuel; Zuazua Iriondo, Enrique (Systems & Control Letters, 2020)
We introduce and study the turnpike property for time-varying shapes, within the viewpoint of optimal control. We focus here on second-order linear parabolic equations where the shape acts as a source term and we seek the optimal time-varying shape that minimizes a quadratic criterion. We first establish existence of optimal solutions under some appropriate sufficient conditions. We then provide necessary conditions for optimality in terms of adjoint equations and, using the concept of strict dissipativity, we prove that state and adjoint satisfy the measure-turnpike property, meaning that the extremal time-varying solution remains essentially close to the optimal solution of an associated static problem. We show that the optimal shape enjoys the exponential turnpike property in terms of Hausdorff distance for a Mayer quadratic cost. We illustrate the turnpike phenomenon in optimal shape design with several numerical simulations.
CRIS-Team Scopus Importer:2020-06-30
Keywords: Direct methods; Optimal shape design; Parabolic equation; Strict dissipativity; Turnpike

Sharp Observability Estimates for Heat Equations - Ervedoza, Sylvain; Zuazua, Enrique (Archive for Rational Mechanics and Analysis, 2011)
The goal of this article is to derive new estimates for the cost of observability of heat equations. We have developed a new method allowing one to show that when the corresponding wave equation is observable, the heat equation is also observable. This method allows one to describe the explicit dependence of the observability constant on the geometry of the problem (the domain in which the heat process evolves and the observation subdomain). We show that our estimate is sharp in some cases, particularly in one space dimension and in the multi-dimensional radially symmetric case. Our result extends those in Fattorini and Russell (Arch Rational Mech Anal 43:272-292, 1971) to the multi-dimensional setting and improves those available in the literature, namely those by Miller (J Differ Equ 204(1):202-226, 2004; SIAM J Control Optim 45(2):762-772, 2006; Atti Accad Naz Lincei Cl Sci Fis Mat Natur Rend Lincei (9) Mat Appl 17(4):351-366, 2006) and Tenenbaum and Tucsnak (J Differ Equ 243(1):70-100, 2007). Our approach is based on an explicit representation formula of some solutions of the wave equation in terms of those of the heat equation, in contrast to the standard application of transmutation methods, which uses a reverse representation of the heat solution in terms of the wave one. We shall also explain how our approach applies and yields some new estimates on the cost of observability in the particular case of the unit square observed from one side. We will also comment on the applications of our techniques to controllability properties of heat-type equations.
CRIS-Team WoS Importer:2019-12-13

Sidewise control of 1-d waves - Saraç, Yeşim; Zuazua Iriondo, Enrique (Journal of Optimization Theory and Applications, 2022)
We analyze the sidewise controllability for the variable coefficients one-dimensional wave equation. The control is acting on one extreme of the string with the aim that the solution tracks a given path or profile at the other free end. This sidewise profile control problem is also often referred to as nodal profile or tracking control. The problem is reformulated as a dual observability property for the corresponding adjoint system, which is proved by means of sidewise energy propagation arguments in a sufficiently large time, in the class of BV-coefficients. We also present a number of open problems and perspectives for further research.<br
Keywords: 1-d wave equations; sidewise profile controllability; nodal profile control; BV-coefficients; sidewise observability; sidewise energy estimates

Sidewise Profile Control of 1-D Waves - Saraç, Yeşim; Zuazua Iriondo, Enrique (Journal of Optimization Theory and Applications, 2022)
We analyze the sidewise controllability for the variable coefficients one-dimensional wave equation. The control is acting on one extreme of the string with the aim that the solution tracks a given path or profile at the other free end. This sidewise profile control problem is also often referred to as nodal profile or tracking control. The problem is reformulated as a dual observability property for the corresponding adjoint system, which is proved by means of sidewise energy propagation arguments in a sufficiently large time, in the class of BV-coefficients. We also present a number of open problems and perspectives for further research.
CRIS-Team Scopus Importer:2022-01-14
Keywords: 1-D wave equations; BV-coefficients; Nodal profile control; Sidewise energy estimates; Sidewise observability; Sidewise profile controllability

SLOW DECAY AND TURNPIKE FOR INFINITE-HORIZON HYPERBOLIC LINEAR QUADRATIC PROBLEMS - Zuazua Iriondo, Enrique; Han, Zhong Jie (SIAM Journal on Control and Optimization, 2022)
This paper is devoted to analyzing the explicit slow decay rate and turnpike in infinite-horizon linear quadratic optimal control problems for hyperbolic systems. Under suitable weak observability or controllability conditions, lower and upper bounds of the corresponding algebraic Riccati operator are proved. Then based on these two bounds, the explicit slow decay rate of the closed-loop system with Riccati-based optimal feedback control is obtained. The averaged turnpike property for this problem is also further discussed. We then apply these results to LQ optimal control problems constrained to networks of one-dimensional wave equations and also some multidimensional ones with local controls which lack a geometric control condition.
CRIS-Team Scopus Importer:2022-09-30
Keywords: observability; optimal control problems; Riccati operator; slow decay rate; turnpike property; weak controllability

Slow decay and Turnpike for Infinite-horizon Hyperbolic LQ problems - Han, Zhong-Jie; Zuazua Iriondo, Enrique (2024)
This paper is devoted to analysing the explicit slow decay rate and turnpike in the infinite-horizon linear quadratic optimal control problems for hyperbolic systems. Assume that some weak observability or controllability are satisfied, by which, the lower and upper bounds of the corresponding algebraic Riccati operator are estimated, respectively. Then based on these two bounds, the explicit slow decay rate of the closed-loop system with Riccati-based optimal feedback control is obtained. The averaged turnpike property for this problem is also further discussed. We then apply these results to the LQ optimal control problems constraint to networks of onedimensional wave equations and also some multi-dimensional ones with local controls which lack of GCC (Geometric Control Condition).
https://cris.fau.de/converis/publicweb/Publication/263168060
Keywords: Optimal control problems; Riccati operator; slow decay rate; weak controllability and observability; turnpike property

Some controllability results for the 2D Kolmogorov equation - Beauchard, K.; Zuazua, Enrique (Annales de l'Institut Henri Poincaré - Analyse Non Linéaire, 2009)
In this article, we prove the null controllability of the 2D Kolmogorov equation both in the whole space and in the square. The control is a source term in the right-hand side of the equation, located on a subdomain, that acts linearly on the state. In the first case, it is the complementary of a strip with axis x and in the second one, it is a strip with axis x.
CRIS-Team WoS Importer:2019-12-13

Sparse initial data identification for parabolic PDE and its finite element approximations - Casas, Eduardo; Vexler, Boris; Zuazua, Enrique (Mathematical Control and Related Fields, 2015)
We address the problem of inverse source identication for parabolic equations from the optimal control viewpoint employing measures of minimal norm as initial data. We adopt the point of view of approximate controllability so that the target is not required to be achieved exactly but only in an approximate sense. We prove an approximate inversion result and derive a characterization of the optimal initial measures by means of duality and the minimization of a suitable quadratic functional on the solutions of the adjoint system. We prove the sparsity of the optimal initial measures showing that they are supported in sets of null Lebesgue measure. As a consequence, approximate controllability can be achieved efficiently by means of controls that are activated in a nite number of pointwise locations. Moreover, we discuss the nite element numerical approximation of the control problem providing a convergence result of the corresponding optimal measures and states as the discretization parameters tend to zero.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Approximate controllability; Borel measures; Parabolic equations; Sparse controls

Sparse source identification of linear diffusion–advection equations by adjoint methods - Monge, Azahar; Zuazua Iriondo, Enrique (Systems & Control Letters, 2020)
We present an algorithm for the time-inversion of diffusion–advection equations, based on the adjoint methodology. Given a final state distribution our main aim is to recover sparse initial conditions, constituted by a finite combination of Kronecker deltas, identifying their location and mass. We discuss the strengths of the adjoint machinery and the difficulties that are to be faced, in particular when the diffusivity coefficient or the time horizon is large.
CRIS-Team Scopus Importer:2020-10-16
Keywords: Adjoint problem; Diffusion–advection equation; Inverse problems; Optimal control; Optimization

Spectral shape optimization for the Neumann traces of the Dirichlet-Laplacian eigenfunctions - Privat, Yannick; Trelat, Emmanuel; Zuazua, Enrique (Calculus of Variations and Partial Differential Equations, 2019)
We consider a spectral optimal design problem involving the Neumann traces of the Dirichlet-Laplacian eigenfunctions on a smooth bounded open subset Ω of [InlineEquation not available: see fulltext.]. The cost functional measures the amount of energy that Dirichlet eigenfunctions concentrate on the boundary and that can be recovered with a bounded density function. We first prove that, assuming a L 1 constraint on densities, the so-called Rellich functions maximize this functional. Motivated by several issues in shape optimization or observation theory where it is relevant to deal with bounded densities, and noticing that the L ∞ -norm of Rellich functions may be large, depending on the shape of Ω , we analyze the effect of adding pointwise constraints when maximizing the same functional. We investigate the optimality of bang–bang functions and Rellich densities for this problem. We also deal with similar issues for a close problem, where the cost functional is replaced by a spectral approximation. Finally, this study is completed by the investigation of particular geometries and is illustrated by several numerical simulations.
CRIS-Team Scopus Importer:2021-06-15

Spike controls for elliptic and parabolic PDEs - Casas, Eduardo; Zuazua, Enrique (Systems & Control Letters, 2013)
We analyze the use of measures of the minimal norm to control elliptic and parabolic equations. We prove the sparsity of the optimal control. In the parabolic case, we prove that the solution of the optimization problem is a Borel measure supported in a set of Lebesgue measure zero. In both cases, the approximate controllability can be achieved efficiently by means of controls that are activated in some finite number of pointwise locations. We also analyze the corresponding dual problem. (C) 2013 Elsevier B.V. All rights reserved.
CRIS-Team WoS Importer:2019-12-13

Stability and Convergence of a Randomized Model Predictive Control Strategy - Veldman, Daniel; Borkowski, Alexandra; Zuazua, Enrique (IEEE Transactions on Automatic Control, 2024)
RBM-MPC is a computationally efficient variant of Model Predictive Control (MPC) in which the Random Batch Method (RBM) is used to speed up the finite-horizon optimal control problems at each iteration. In this paper, stability and convergence estimates are derived for RBM-MPC of unconstrained linear systems. The obtained estimates are validated in a numerical example that also shows a clear computational advantage of RBM-MPC.
CRIS-Team Scopus Importer:2024-03-22
Keywords: Approximation algorithms; Computational efficiency; Convergence; Error Estimates; Model Predictive Control; Numerical stability; Random Batch Method; Read only memory; Receding Horizon Control; Stability; Stability criteria; Vectors

Stability and Convergence of a Randomized Model Predictive Control Strategy - Veldman, Daniel; Borkowski, Alexandra; Zuazua Iriondo, Enrique (2025)
This paper is concerned with a combination of Random Batch Methods (RBMs) and Model Predictive Control (MPC) called RBM-MPC. In RBM-MPC, the RBM is used to speed up the solution of the finite horizon optimal control problems that need to be solved in MPC. We analyze our algorithm in the linear quadratic setting and obtain explicit error estimates that characterize the stability and convergence of the proposed method. The obtained estimates are validated in numerical experiments that also demonstrate the effectiveness of RBM-MPC.<br
https://cris.fau.de/converis/publicweb/Publication/285034893
Keywords: Convergence; Model Predictive Control; Random Batch Method; Receding Horizon Control; Stability

Stabilization of the wave equation on 1-D networks - Valein, Julie; Zuazua, Enrique (SIAM Journal on Control and Optimization, 2009)
In this paper we study the stabilization of the wa ve equation on general 1-d networks. For that, we transfer known observability results in the context of control problems of conservative systems (see [R. Dáger and E. Zuazua, Wave Propagation, Observation, and Control in 1-d Flexible Multi-structures, Math. Appl. 50, Springer-Verlag, Berlin, 2006]) into a weighted observability estimate for dissipative systems. Then we use an interpolation inequality similar to the one proved in [P. Bégout and F. Soria, J. Differential Equations, 240 (2007), pp. 324-356] to obtain the explicit decay estimates of the energy for smooth initial data. The obtained decay rate depends on the geometric and topological properties of the network. We also give some examples of particular networks in which our results apply, yielding different decay rates.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Network; Stabilization; Wave equation

Stable observation of additive superpositions of Partial Differential Equations - Zuazua, Enrique (Systems & Control Letters, 2016)
We address the problem of observing the solutions of a given Partial Differential Equation (PDE), submitted to unknown additive perturbations fulfilling another PDE, possibly of different nature. The problem under consideration is motivated by the issue of averaged control and observation. In the latter, one aims at determining the initial datum of a dynamics, submitted to unknown perturbations, by means of measurements done on the average of solutions with respect to the unknown parameters. We do it in two different cases. First, we assume that a given dynamics is perturbed in an additive manner by a state governed by a PDE, but not necessarily of the same type. In particular we consider the additive superposition of multi-physics solutions of wave and heat type, for instance. Assuming the PDEs under consideration have constant coefficients, we prove that, the by now well-known observability properties of PDE (mainly heat and wave equations) from open subsets of the domain where the dynamics evolves, are stable under these additive perturbations. The proof is rather systematic and easy to apply. It consists in composing all but one (the observable one) PDE operators so to reduce the problem to the consideration of a single equation and applying its known observability properties. The same proof leads to very interesting open problems for systems involving PDEs with variable coefficients where commutators lead to non-local lower order perturbations. We then consider a particular case of a PDE depending on a parameter in a continuous manner. We show that the average is governed by the superposition of two distinct solutions, corresponding to the extremal values of the parameter, and experiencing an extra regularizing effect. This allows showing that the averaged observability property holds, for the continuous average, but to the price of losing a finite number of derivatives. This opens a wide range of interesting problems to be explored.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Additive superposition; Averages; Controllability; Parameter dependent Partial Differential Equations

Steady-state and periodic exponential turnpike property for optimal control problems in hilbert spaces - Trelat, Emmanuel; Zhang, Can; Zuazua, Enrique (SIAM Journal on Control and Optimization, 2018)
In this work, we study the steady-state (or periodic) exponential turnpike property of optimal control problems in Hilbert spaces. The turnpike property, which is essentially due to the hyperbolic feature of the Hamiltonian system resulting from the Pontryagin maximum principle, reects the fact that, in large control time horizons, the optimal state and control and adjoint state remain most of the time close to an optimal steady-state. A similar statement holds true as well when replacing an optimal steady-state by an optimal periodic trajectory. To establish the result, we design an appropriate dichotomy transformation, based on solutions of the algebraic Riccati and Lyapunov equations. We illustrate our results with examples including linear heat and wave equations with periodic tracking terms.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Dichotomy transformation; Exponential turnpike property; Periodic optimal controls; Periodic tracking; Stability analysis

Switching control - Zuazua, Enrique (Journal of the European Mathematical Society, 2011)
We analyze the problem of switching controls for control systems endowed with different actuators. The goal is to control the dynamics of the system by switching from an actuator to another in a systematic way so that, at each instant of time, only one actuator is active. We first address a finite-dimensional model and show that, under suitable rank conditions, switching control strategies exist and can be built in a systematic way. To do this we introduce a new variational principle building a new functional based on the adjoint system whose minimizers yield the switching controls. When the above rank condition fails, the same variational strategy applies but the controls obtained this way fail to be of switching form since they may be, for some instants of time, convex combinations of both controllers. We then address the same issue for the 1-d heat equation endowed with two pointwise controls. We show that, due to the time analyticity of solutions, under suitable conditions on the location of the controllers, switching control strategies exist. We also show that the controls we obtain are optimal in the sense that, for instance, for two scalar valued controls, they are of minimal L-2(0, T ; R-2)-norm, the space R-2 being endowed with the l(1)-norm. We also discuss some possible extensions to multi-dimensional heat equations which require a preliminary analysis of generic properties of the spectrum that, as far as we know, are not yet well understood.
CRIS-Team WoS Importer:2019-12-13

The asymptotic behaviour of the heat equation in a twisted Dirichlet-Neumann waveguide - Krejcirik, David; Zuazua, Enrique (Journal of Differential Equations, 2011)
We consider the heat equation in a straight strip, subject to a combination of Dirichlet and Neumann boundary conditions. We show that a switch of the respective boundary conditions leads to an improvement of the decay rate of the heat semigroup of the order of t(-1/2). The proof employs similarity variables that lead to a non-autonomous parabolic equation in a thin strip contracting to the real line, that can be analysed on weighted Sobolev spaces in which the operators under consideration have discrete spectra. A careful analysis of its asymptotic behaviour shows that an added Dirichlet boundary condition emerges asymptotically at the switching point, breaking the real line in two half-lines, which leads asymptotically to the 1/2 gain on the spectral lower bound, and the t(-1/2) gain on the decay rate in the original physical variables. This result is an adaptation to the case of strips with twisted boundary conditions of previous results by the authors on geometrically twisted Dirichlet tubes. (C) 2010 Elsevier Inc. All rights reserved.
CRIS-Team WoS Importer:2019-12-13

The Finite-Time Turnpike Phenomenonfor Optimal Control Problems:Stabilization by Non-smooth TrackingTerms - Gugat, Martin; Zuazua Iriondo, Enrique; Schuster, Michael (Stabilization of Distributed Parameter Systems: Design Methods and Applications, 2021)
In this paper, problems of optimal control are considered where in the objective function, in addition to the control cost there is a tracking term that measures the distance to a desired stationary state. The tracking term is given by some norm and therefore it is in general not differentiable. In the optimal control problem, the initial state is prescribed. We assume that the system is either exactly controllable in the classical sense or nodal profile controllable. We show that both for systems that are governed by ordinary differential equations and for infinite-dimensional systems, for example for boundary control systems governed by the wave equation, under certain assumptions the optimal system state is steered exactly to the desired state after finite tim
Keywords: Turnpike phenomenon; finite time; optimal control problems

The Hardy inequality and the heat equation in twisted tubes - Krejcirik, David; Zuazua, Enrique (Journal De Mathematiques Pures Et Appliquees, 2010)
We show that a twist of a three-dimensional tube of uniform cross-section yields an improved decay rate for the heat semigroup associated with the Dirichlet Laplacian in the tube. The proof employs Hardy inequalities for the Dirichlet Laplacian in twisted tubes and the method of self-similar variables and weighted Sobolev spaces for the heat equation. (C) 2010 Elsevier Masson SAS. All rights reserved.
CRIS-Team WoS Importer:2019-12-13

The inverse problem for Hamilton-jacobi equations and semiconcave envelopes - Esteve, Carlos; Zuazua Iriondo, Enrique (SIAM Journal on Mathematical Analysis, 2020)
We study the inverse problem, or inverse design problem, for a time-evolution Hamilton-Jacobi equation. More precisely, given a target function uT and a time horizon T > 0, we aim to construct all the initial conditions for which the viscosity solution coincides with uT at time T. As is common in this kind of nonlinear equation, the target might not be reachable. We first study the existence of at least one initial condition leading the system to the given target. The natural candidate, which indeed allows determining the reachability of uT , is the one obtained by reversing the direction of time in the equation, considering uT as terminal condition. In this case, we use the notion of backward viscosity solution, which provides existence and uniqueness for the terminal-value problem. We also give an equivalent reachability condition based on a differential inequality, which relates the reachability of the target with its semiconcavity properties. Then, for the case when uT is reachable, we construct the set of all the initial conditions for which the viscosity solution coincides with uT at time T. Note that, in general, such initial conditions are not unique. Finally, for the case when the target uT is not necessarily reachable, we study the projection of uT on the set of reachable targets, obtained by solving the problem backward and then forward in time. This projection is then identified with the solution of a fully nonlinear obstacle problem and can be interpreted as the semiconcave envelope of uT , i.e., the smallest reachable target bounded from below by uT.
CRIS-Team Scopus Importer:2020-12-11
Keywords: Hamilton-jacobi equation; Inverse design problem; Obstacle problems; Semiconcave envelopes

The inverse problem for Hamilton-Jacobi equations and semiconcave envelopes - Esteve Yagüe, Carlos; Zuazua Iriondo, Enrique (Workshop on Mathematical Analysis, 2020)
We study the inverse problem, or inverse design problem, for a time-evolution Hamilton-Jacobi equation. More precisely, given a target function u_T and a time horizon T>0, we aim to construct all the initial conditions for which the viscosity solution coincides with u_T at time T. As it is common in this kind of nonlinear equations, the target might not be reachable. We first study the existence of at least one initial condition leading the system to the given target. The natural candidate, which indeed allows determining the reachability of u_T, is the one obtained by reversing the direction of time in the equation, considering u_T as terminal condition. In this case, we use the notion of backward viscosity solution, that provides existence and uniqueness for the terminal-value problem. We also give an equivalent reachability condition based on a differential inequality, that relates the reachability of the target with its semiconcavity properties. Then, for the case when u_T is reachable, we construct the set of all initial conditions for which the solution coincides with u_T at time T. Note that in general, such initial conditions are not unique. Finally, for the case when the target u_T is not necessarily reachable, we study the projection of u_T on the set of reachable targets, obtained by solving the problem backward and then forward in time. This projection is then identified with the solution of a fully nonlinear obstacle problem, and can be interpreted as the semiconcave envelope of u_T, i.e. the smallest reachable target bounded from below by u_T.<br

The turnpike property and the longtime behavior of the Hamilton-Jacobi-Bellman equation for finite-dimensional LQ control problems - Esteve Yagüe, Carlos; Kouhkouh, Hicham; Pighin, Dario; Zuazua Iriondo, Enrique (Mathematics of Control Signals and Systems, 2022)
We analyze the consequences that the so-called turnpike property has on the long-time behavior of the value function corresponding to a finite-dimensional linear-quadratic optimal control problem with general terminal cost and constrained controls. We prove that, when the time horizon T tends to infinity, the value function asymptotically behaves as [katex]W(x)+cT+λ [/katex], , and we provide a control interpretation of each of these three terms, making clear the link with the turnpike property. As a by-product, we obtain the long-time behavior of the solution to the associated Hamilton-Jacobi-Bellman equation in a case where the Hamiltonian is not coercive in the momentum variable. As a result of independent interest, we showed that linear-quadratic optimal control problems with constrained control enjoy a turnpike property, also particularly when the steady optimum may saturate the control constr

The turnpike property in finite-dimensional nonlinear optimal control - Trelat, Emmanuel; Zuazua, Enrique (Journal of Differential Equations, 2015)
Turnpike properties have been established long time ago in finite-dimensional optimal control problems arising in econometry. They refer to the fact that, under quite general assumptions, the optimal solutions of a given optimal control problem settled in large time consist approximately of three pieces, the first and the last of which being transient short-time arcs, and the middle piece being a long-time arc staying exponentially close to the optimal steady-state solution of an associated static optimal control problem. We provide in this paper a general version of a turnpike theorem, valuable for nonlinear dynamics without any specific assumption, and for very general terminal conditions. Not only the optimal trajectory is shown to remain exponentially close to a steady-state, but also the corresponding adjoint vector of the Pontryagin maximum principle. The exponential closedness is quantified with the use of appropriate normal forms of Riccati equations. We show then how the property on the adjoint vector can be adequately used in order to initialize successfully a numerical direct method, or a shooting method. In particular, we provide an appropriate variant of the usual shooting method in which we initialize the adjoint vector, not at the initial time, but at the middle of the trajectory.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Direct methods; Optimal control; Pontryagin maximum principle; Riccati equation; Shooting method; Turnpike

The turnpike property in nonlinear optimal control - A geometric approach - Sakamoto, Noboru; Pighin, Dario; Zuazua, Enrique (Proceedings of the IEEE Conference on Decision and Control, 2019)
This paper presents, using dynamical system theory, a framework for investigating the turnpike property in nonlinear optimal control. First, it is shown that a turnpike-like property appears in general dynamical systems with hyperbolic equilibrium and then, apply it to optimal control problems to obtain sufficient conditions for the turnpike occurs. The approach taken is geometric and gives insights for the behaviors of controlled trajectories, allowing us to find simpler proofs for existing results on the turnpike properties.
CRIS-Team Scopus Importer:2021-06-15

The turnpike property in nonlinear optimal control – A geometric approach - Sakamoto, Noboru; Zuazua Iriondo, Enrique (Automatica, 2021)
This paper presents, using dynamical system theory, a framework for investigating the turnpike property in nonlinear optimal control. First, it is shown that a turnpike-like property appears in general dynamical systems with hyperbolic equilibrium and then, apply it to optimal control problems to obtain sufficient conditions for the turnpike to occurs. The approach taken is geometric and gives insights for the behaviors of controlled trajectories as well as links between the turnpike and stability and/or stabilizability in nonlinear control theory. It also allows us to find simpler proofs for existing results on the turnpike properties. Attempts to remove smallness restrictions for initial and target states are also discussed based on the geometry of (un)stable manifolds and a recent result on exponential stabilizability of nonlinear control systems obtained by one of the authors.<br
Keywords: Nonlinear system; optimal control; Turnpike

The turnpike property in nonlinear optimal control-A geometric approach - Sakamoto, Noboru; Zuazua Iriondo, Enrique (Automatica, 2021)
This paper presents, using dynamical system theory, a framework for investigating the turnpike property in nonlinear optimal control. First, it is shown that a turnpike-like property appears in general dynamical systems with hyperbolic equilibrium and then, apply it to optimal control problems to obtain sufficient conditions for the turnpike behavior to occur. The approach taken is geometric and gives insights for the behaviors of controlled trajectories as well as links between the turnpike property and stability and/or stabilizability in nonlinear control theory. It also allows us to find simpler proofs for existing results on the turnpike properties. Attempts to remove smallness restrictions for initial and target states are also discussed based on the geometry of (un)stable manifolds and a recent result on exponential stabilizability of nonlinear control systems obtained by one of the authors. (C) 2021 Elsevier Ltd. All rights reserved.
CRIS-Team WoS Importer:2021-10-29

The Vlasov-Fokker-Planck equation with high dimensional parametric forcing term - Jin, Shi; Zhu, Yuhua; Zuazua Iriondo, Enrique (Numerische Mathematik, 2022)
We consider the Vlasov-Fokker-Planck equation with random electric field where the random field is parametrized by countably many infinite random variables due to uncertainty. At the theoretical level, with suitable assumption on the anisotropy of the randomness, adopting the technique employed in elliptic PDEs (Cohen and DeVore in Acta Numerica 24:1-159, 2015) , we prove the best N approximation in the random space enjoys a convergence rate, which depends on the summability of the coefficients of the random variable, higher than the Monte-Carlo method. For the numerical method, based on the adaptive sparse polynomial interpolation (ASPI) method introduced in Chkifa et al. (Found Comput Math 14:601-603, 2014), we develop a residual based adaptive sparse polynomial interpolation (RASPI) method which is more efficient for multi-scale linear kinetic equation, when using numerical schemes that are time dependent and implicit. Numerical experiments show that the numerical error of the RASPI decays faster than the Monte-Carlo method and is also dimension independent.
CRIS-Team WoS Importer:2022-01-28

The Vlasov-Fokker-Planck Equation with High Dimensional Parametric Forcing Term - Jin, Shi; Zhu, Yuhua; Zuazua Iriondo, Enrique (Numerische Mathematik, 2022)
We consider the Vlasov-Fokker-Planck equation with random electric field where the random field is parametrized by countably many infinite random variables due to uncertainty. At the theoretical level, with suitable assumption on the anisotropy of the randomness, adopting the technique employed in elliptic PDEs [5], we prove the best N approximation in the random space breaks the dimension curse and the convergence rate is faster than the Monte Carlo method. For the numerical method, based on the adaptive sparse polynomial interpolation (ASPI) method introduced in [2], we develop a residual based adaptive sparse polynomial interpolation (RASPI) method which is more efficient for multi-scale linear kinetic equation, when using numerical schemes that are time dependent and implicit. Numerical experiments show that the numerical error of the RASPI decays faster than the Monte-Carlo method and is also dimension independent<br
Keywords: Dimension curse; Kinetic equation; Fokker-Planck operator; Hypocoercivity; Adaptive sparse polynomial interpolation; Residual based greedy algorithm

The Wave Equation: Control and Numerics - Ervedoza, Sylvain; Zuazua, Enrique (Lecture Notes in Mathematics, 2012)
In these Notes we make a self-contained presentation of the theory that has been developed recently for the numerical analysis of the controllability properties of wave propagation phenomena and, in particular, for the constant coefficient wave equation. We develop the so-called discrete approach. In other words, we analyze to which extent the semidiscrete or fully discrete dynamics arising when discretizing the wave equation by means of the most classical scheme of numerical analysis, shear the property of being controllable, uniformly with respect to the mesh-size parameters and if the corresponding controls converge to the continuous ones as the mesh-size tends to zero. We focus mainly on finite-difference approximation schemes for the one-dimensional constant coefficient wave equation. Using the well known equivalence of the control problem with the observation one, we analyze carefully the second one, which consists in determining the total energy of solutions out of partial measurements. We show how spectral analysis and the theory of non-harmonic Fourier series allows, first, to show that high frequency wave packets may behave in a pathological manner and, second, to design efficient filtering mechanisms. We also develop the multiplier approach that allows to provide energy identities relating the total energy of solutions and the energy concentrated on the boundary. These observability properties obtained after filtering, by duality, allow to build controls that, normally, do not control the full dynamics of the system but rather guarantee a relaxed controllability property. Despite of this they converge to the continuous ones. We also present a minor variant of the classical Hilbert Uniqueness Method allowing to build smooth controls for smooth data. This result plays a key role in the proof of the convergence rates of the discrete controls towards the continuous ones. These results are illustrated by means of several numerical experiments.
CRIS-Team WoS Importer:2019-12-13

TIME DISCRETE WAVE EQUATIONS: BOUNDARY OBSERVABILITY AND CONTROL - Zhang, Xu; Zheng, Chuang; Zuazua, Enrique (Discrete and Continuous Dynamical Systems, 2009)
In this paper we study the exact boundary controllability of a trapezoidal time discrete wave equation in a bounded domain. We prove that the projection of the solution in an appropriate filtered space is exactly controllable with uniformly bounded cost with respect to the time-step. In this way, the well-known exact-controllability property of the wave equation can be reproduced as the limit, as the time step h -> 0, of the controllability of projections of the time-discrete one. By duality these results are equivalent to deriving uniform observability estimates (with respect to h -> 0) within a class of solutions of the time-discrete problem in which the high frequency components have been filtered. The later is established by means of a time-discrete version of the classical multiplier technique. The optimality of the order of the filtering parameter is also established, although a careful analysis of the expected velocity of propagation of time-discrete waves indicates that its actual value could be improved.
CRIS-Team WoS Importer:2019-12-13

Tracking controllability for finite-dimensional linear systems - Zamorano Aliaga, Sebastian Andres; Zuazua Iriondo, Enrique (2024)
We introduce and analyze the concept of tracking controllability, where the objective is for the state of a control system, or some projection thereof, to follow a given trajectory within a specified time horizon. By leveraging duality, we characterize tracking controllability through a non-standard observability inequality for the adjoint system. This characterization facilitates the development of a systematic approach for constructing controls with minimal norm. However, achieving such an observability property is challenging. To address this, we utilize the classical Brunovský canonical form for control systems, specifically focusing on the case of scalar controls. Our findings reveal that tracking control inherently involves a loss of regularity in the controls. This phenomenon is highly sensitive to the system's structure and the interaction between the projection to be tracked and the system.<br
https://cris.fau.de/converis/publicweb/Publication/326481202

Tracking Controllability for the Heat Equation - Barcena Petisco, Jon Asier; Zuazua Iriondo, Enrique (IEEETransactions on Automatic Control, 2024)
We study the tracking or sidewise controllability of the heat equation. More precisely, we seek for controls that, acting on part of the boundary of the domain where the heat process evolves, aim to assure that the normal trace or flux on the complementary set tracks a given trajectory. The dual equivalent observability problem is identified. It consists on estimating the boundary sources, localized on a given subset of the boundary, out of boundary measurements on the complementary subset. Classical unique continuation and smoothing properties of the heat equation allow us proving approximate tracking controllability properties and the smoothness of the class of trackable trajectories. We also develop a new transmutation method which allows to transfer known results on the sidewise controllability of the wave equation to the tracking controllability of the heat one. Using the flatness approach we also give explicit estimates on the cost of approximate tracking control. The analysis is complemented with a discussion of some possible variants of these results and a list of open problems.
CRIS-Team Scopus Importer:2024-11-01
Keywords: Linear system observers; linear systems; optimal control; tracking controllability

Tracking control of 1D scalar conservation laws in the presence of shocks - Lecaros, Rodrigo; Zuazua, Enrique (Trends in Contemporary Mathematics, 2014)
We analyze a model tracking problem for a 1D scalar conservation law. It consists in optimizing the initial datum so to minimize a weighted distance to a given target during a given finite time horizon. Even if the optimal control problem under consideration is of classical nature, the presence of shocks is an impediment for classical methods, based on linearization, to be directly applied. We adapt the so-called alternating descent method that exploits the generalized linearization that takes into account both the sensitivity of the shock location and of the smooth components of solutions. This method was previously applied successfully on the inverse design problem and that of identifying the nonlinearity in the equation. The efficiency of the method in comparison with more classical discrete methods is illustrated by several numerical experiments.
CRIS-Team Scopus Importer:2021-06-15

Transmutation techniques and observability for time-discrete approximation schemes of conservative systems - Ervedoza, Sylvain; Zuazua, Enrique (Numerische Mathematik, 2015)
In this article, we consider abstract linear conservative systems and their time-discrete counterparts. Our main result is a representation formula expressing solutions of the continuous model through the solution of the corresponding time-discrete one. As an application, we show how observability properties for the time continuous model yield uniform (with respect to the time-step) observability results for its time-discrete approximation counterparts, provided the initial data are suitably filtered. The main output of this approach is the estimate on the time under which we can guarantee uniform observability for the time-discrete models. Besides, using a reverse representation formula, we also prove that this estimate on the time of uniform observability for the time-discrete models is sharp. We then conclude with some general comments and open problems.
CRIS-Team Scopus Importer:2021-06-15
Keywords: 35C15; 35L90; 65J08; 93B07; 93B17; 93B40

Turnpike in Control of PDEs, RESNETs and beyond - Geshkovski, Borjan; Zuazua Iriondo, Enrique (2024)
The turnpike property in contemporary macroeconomics asserts that if an economic planner seeks to move an economy from one level of capital to another, then the most efficient path, as long as the planner has enough time, is to rapidly move stock to a level close to the optimal stationary or constant path, then allow for capital to develop along that path until the desired term is nearly reached, at which point the stock ought to be moved to the final target. Motivated in part by its nature as a resource allocation strategy, over the past decade, the turnpike property has also been shown to hold for several classes of partial differential equations arising in mechanics. When formalized math- ematically, the turnpike theory corroborates the insights from economics: for an optimal control problem set in a finite-time horizon, optimal controls and corresponding states, are close (often exponentially), during most of the time, except near the initial and final time, to the optimal control and corresponding state for the associated stationary optimal control problem. In particular, the former are mostly constant over time. This fact provides a rigorous meaning to the asymptotic simplification that some optimal control problems appear to enjoy over long time intervals, allowing the consideration of the corre- sponding stationary problem for computing and applications. We review a slice of the theory developed over the past decade –the controllability of the underlying system is an important ingredient, and can even be used to de- vise simple turnpike-like strategies which are nearly optimal–, and present several novel applications, including, among many others, the characteriza- tion of Hamilton-Jacobi-Bellman asymptotics, and stability estimates in deep learning via residual neural networks.<br
https://cris.fau.de/converis/publicweb/Publication/269395963
Keywords: Turnpike property; Partial Differential Equations; Optimal control; Shape optimization; Machine learning

Turnpike in Lipschitz-nonlinear optimal control - Esteve Yagüe, Carlos; Geshkovski, Borjan; Pighin, Dario; Zuazua Iriondo, Enrique (Nonlinearity, 2022)
We present a new proof of the turnpike property for nonlinear optimal control problems, when the running target is a steady control-state pair of the underlying dynamics. Our strategy combines the construction of suboptimal quasi-turnpike trajectories via controllability, and a bootstrap argument, and does not rely on analyzing the optimality system or linearization techniques. This in turn allows us to address several optimal control problems for finite-dimensional, control-affine systems with globally Lipschitz (possibly nonsmooth) nonlinearities, without any smallness conditions on the initial data or the running target. These results are motivated by the large-layer regime of residual neural networks, commonly used in deep learning applications. We show that our methodology is applicable to controlled PDEs as well, such as the semilinear wave and heat equation with a globally Lipschitz nonlinearity, once again without any smallness assumptions.
Keywords: Optimal control; Turnpike property; Neural ODEs; ResNets; Deep learning; Wave equation; Heat equation

Turnpike in optimal control and beyond: a survey - Trélat, Emmanuel; Zuazua Iriondo, Enrique (2025)
The turnpike principle is a fundamental concept in optimal control theory, stating that for a wide class of long-horizon optimal control problems, the optimal trajectory spends most of its time near a steady-state solution (the “turnpike”) rather than being influenced by the initial or final conditions. In this article, we provide a survey on the turnpike property in optimal control, adding several recent and novel considerations. After some historical insights, we present an elemen- tary proof of the exponential turnpike property for linear-quadratic optimal control problems in finite dimension. Next, we show an extension to nonlinear optimal control problems, with a local exponential turnpike property. On simple but meaningful examples, we illustrate the local and global aspects of the turnpike theory, clarifying the global picture and raising new questions. We discuss key generalizations, in infinite dimension and other various settings, and review several applications of the turnpike theory across different fields.
https://cris.fau.de/converis/publicweb/Publication/338490048

Turnpike in optimal control of PDEs, ResNets, and beyond - Geshkovski, Borjan; Zuazua Iriondo, Enrique (Acta Numerica, 2022)
The turnpike property in contemporary macroeconomics asserts that if an economic planner seeks to move an economy from one level of capital to another, then the most efficient path, as long as the planner has enough time, is to rapidly move stock to a level close to the optimal stationary or constant path, then allow for capital to develop along that path until the desired term is nearly reached, at which point the stock ought to be moved to the final target. Motivated in part by its nature as a resource allocation strategy, over the past decade, the turnpike property has also been shown to hold for several classes of partial differential equations arising in mechanics. When formalized mathematically, the turnpike theory corroborates insights from economics: for an optimal control problem set in a finite-time horizon, optimal controls and corresponding states are close (often exponentially) most of the time, except near the initial and final times, to the optimal control and the corresponding state for the associated stationary optimal control problem. In particular, the former are mostly constant over time. This fact provides a rigorous meaning to the asymptotic simplification that some optimal control problems appear to enjoy over long time intervals, allowing the consideration of the corresponding stationary problem for computing and applications. We review a slice of the theory developed over the past decade – the controllability of the underlying system is an important ingredient, and can even be used to devise simple turnpike-like strategies which are nearly optimal – and present several novel applications, including, among many others, the characterization of Hamilton–Jacobi–Bellman asymptotics, and stability estimates in deep learning via residual neural networks.<br

Turnpike in optimal shape design - Lance, G.; Trelat, Emmanuel; Zuazua, Enrique (IFAC-PapersOnLine, 2019)
We investigate the turnpike problem in optimal control, in the context of time-evolving shapes. We focus here on the heat equation model where the shape acts as a source term, and we search the optimal time-varying shape, minimizing a quadratic criterion. We first establish existence of optimal solutions under some appropriate sufficient conditions. We provide necessary conditions for optimality in terms of usual adjoint equations and then, thanks to strict dissipativity properties, we prove that state and adjoint satisfy a measure-turnpike property, meaning that the extremal time-varying solution remains essentially close to an optimal solution of an associated static problem. We illustrate the turnpike phenomenon in shape design with several numerical simulations.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Direct methods; Heat equation; Optimal shape design; Strict dissipativity; Turnpike

Uniform exponential decay for viscous damped systems* - Ervedoza, Sylvain; Zuazua, Enrique (Progress in Nonlinear Differential Equations and Their Application, 2009)
We consider a class of viscous damped vibrating systems. We prove that, under the assumption that the damping term ensures the exponential decay for the corresponding inviscid system, then the exponential decay rate is uniform for the viscous one, regardless what the value of the viscosity parameter is. Our method is mainly based on a decoupling argument of low and high frequencies. Low frequencies can be dealt with because of the effectiveness of the damping term in the inviscid case while the dissipativity of the viscous term guarantees the decay of the high-frequency components. This method is inspired in previous work by the authors on time-discretization schemes for damped systems in which a numerical viscosity term needs to be added to ensure the uniform exponential decay with respect to the time-step parameter.
CRIS-Team Scopus Importer:2021-06-15
Keywords: Dissipative semigroups; Elasticity; Exponential decay; Viscoelasticity

Uniformly exponentially stable approximations for a class of damped systems - Ervedoza, Sylvain; Zuazua, Enrique (Journal De Mathematiques Pures Et Appliquees, 2009)
We consider time semi-discrete approximations of a class of exponentially stable infinite-dimensional systems modeling, for instance, damped vibrations. It has recently been proved that for time semi-discrete systems, due to high frequency spurious components, the exponential decay property may be lost as the time step tends to zero. We prove that adding a suitable numerical viscosity term in the numerical scheme, one obtains approximations that are uniformly exponentially stable. This result is then combined with previous ones on space semi -di scretizati ons to derive similar results on fully-discrete approximation schemes. Our method is mainly based on a decoupling argument of low and high frequencies, the low frequency observability property for time semi-discrete approximations of conservative linear systems and the dissipativity of the numerical viscosity on the high frequency components. Our methods also allow to deal directly with stabilization properties of fully discrete approximation schemes without numerical viscosity, under a Suitable CFL type condition on the time and space discretization parameters. (c) 2008 Elsevier Masson SAS. All tights reserved.
CRIS-Team WoS Importer:2019-12-13

Uniform Turnpike Property and Singular Limits - Hernández Salinas, Martin; Zuazua Iriondo, Enrique (Acta Applicandae Mathematicae, 2024)
Motivated by singular limits for long-time optimal control problems, we investigate a class of parameter-dependent parabolic equations. First, we prove a turnpike result, uniform with respect to the parameters within a suitable regularity class and under appropriate bounds. The main ingredient of our proof is the justification of the uniform exponential stabilization of the corresponding Riccati equations, which is derived from the uniform null control properties of the model. Then, we focus on a heat equation with rapidly oscillating coefficients. In the one-dimensional setting, we obtain a uniform turnpike property with respect to the highly oscillatory heterogeneous medium. Afterward, we establish the homogenization of the turnpike property. Finally, our results are validated by numerical experiment
Keywords: Turnpike property; Rapidly oscillating linear parabolic equation; Optimal control problems; Long time behavior; Uniform controllability; Singular limits problems.

Universal Approximation of Dynamical Systems by Semi-Autonomous Neural ODEs and Applications - Li, Ziqian; Liu, Kang; Liverani, Lorenzo; Zuazua Iriondo, Enrique (2025)
In this paper, we introduce semi-autonomous neural ordinary differential equations (SA-NODEs), a variation of the vanilla NODEs, employing fewer parameters. We investigate the universal approximation properties of SA-NODEs for dynamical systems from both a theoretical and a numerical perspective. Within the assumption of a finite-time horizon, under general hypotheses we establish an asymptotic approximation result, demonstrating that the error vanishes as the number of parameters goes to infinity. Under additional regularity assumptions, we further specify this convergence rate in relation to the number of parameters, utilizing quantitative approximation results in the Barron space. Based on the previous result, we prove an approximation rate for transport equations by their neural counterparts. Our numerical experiments validate the effectiveness of SA-NODEs in capturing the dynamics of various ODE systems and transport equations. Additionally, we compare SA-NODEs with vanilla NODEs, highlighting the superior performance and reduced complexity of our approach.<br
https://cris.fau.de/converis/publicweb/Publication/326007759
Keywords: neural ODEs; universal approximation; Barron space; transport equations.

Weak observability estimates for 1-D wave equations with rough coefficients - Fanelli, Francesco; Zuazua, Enrique (Annales de l'Institut Henri Poincaré - Analyse Non Linéaire, 2015)
In this paper we prove observability estimates for 1-dimensional wave equations with non-Lipschitz coefficients. For coefficients in the Zygmund class we prove a "classical" estimate, which extends the well-known observability results in the energy space for BV regularity. When the coefficients are instead log-Lipschitz or log-Zygmund, we prove observability estimates "with loss of derivatives": in order to estimate the total energy of the solutions, we need measurements on some higher order Sobolev norms at the boundary. This last result represents the intermediate step between the Lipschitz (or Zygmund) case, when observability estimates hold in the energy space, and the Hölder one, when they fail at any finite order (as proved in [9]) due to an infinite loss of derivatives. We also establish a sharp relation between the modulus of continuity of the coefficients and the loss of derivatives in the observability estimates. In particular, we will show that under any condition which is weaker than the log-Lipschitz one (not only Hölder, for instance), observability estimates fail in general, while in the intermediate instance between the Lipschitz and the log-Lipschitz ones they can hold only admitting a loss of a finite number of derivatives. This classification has an exact counterpart when considering also the second variation of the coefficients.
CRIS-Team Scopus Importer:2021-06-15
Keywords: 1-D wave equation; Loss of derivatives; Non-Lipschitz coefficients; Observability estimates; Zygmund regularity