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.

},
author = {Veldman, Daniel and Zuazua Iriondo, Enrique},
doi = {10.1007/s00211-022-01290-3},
faupublication = {yes},
journal = {Numerische Mathematik},
keywords = {Random Batch Method; Operator Splitting; Optimal Control; Model Predictive Control},
pages = {495–549},
peerreviewed = {Yes},
title = {{A} framework for randomized time-splitting in linear-quadratic optimal control},
url = {https://link.springer.com/article/10.1007/s00211-022-01290-3},
volume = {151},
year = {2022}
}
@article{faucris.314060134,
abstract = {We deal with the problem of approximating a scalar conservation law by a conservation law with nonlocal flux. As convolution kernel in the nonlocal flux, we consider an exponential-type approximation of the Dirac distribution. We then obtain a total variation bound on the nonlocal term and can prove that the (unique) weak solution of the nonlocal problem converges strongly in C.LIn this paper we look at a new aspect of the turnpike phenomenon. We show that for problems without explicit terminal condition, for large time horizons in the last part of the time interval the optimal state approaches a certain limit trajectory that is independent of the terminal time exponentially fast. For large time horizons also the optimal state in the initial part of the time interval approaches exponentially fast a limit state.

<}, author = {Gugat, Martin and Sokolowski, Jan}, doi = {10.55630/serdica.2023.49.127-154}, faupublication = {yes}, journal = {Serdica Mathematical Journal}, keywords = {turnpike property, linear-quadratic optimal control problem, linear systems, dynamic optimal control problem}, pages = {127-154}, peerreviewed = {Yes}, title = {{An} aspect of the turnpike property. {Long} time horizon behavior}, url = {https://serdica.math.bas.bg/index.php/serdica/article/view/39}, volume = {49}, year = {2023} } @article{faucris.274937530, abstract = {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.}, author = {Huang, Shiping and Zhang, Huijian and Chen, Piaohua and Zhu, Yazhi and Zuazua, Enrique}, doi = {10.12989/sem.2022.82.1.041}, faupublication = {yes}, journal = {Structural Engineering and Mechanics}, keywords = {Beam bridge; Bridge vibration; Multiple constraints; Natural frequency; Vibration mode}, note = {CRIS-Team Scopus Importer:2022-05-13}, pages = {41-53}, peerreviewed = {Yes}, title = {{An} energy-based vibration model for beam bridges with multiple constraints}, volume = {82}, year = {2022} } @incollection{faucris.237312025, abstract = {We consider the flow of gas through networks of pipelines. A hierarchy of models for the gasflow is available. The most accurate model is the pde system given by the 1-d Euler

equations. For large-scale optimization problems, simplifications of this model are

necessary. Here we propose a new model that is derived for high-pressure flows that are

close to stationary flows. For such flows, we can make the assumption of constant gas

velocity. Under this assumption, we obtain a model that allows transient gas flow rates.

0 and Omega(0) subset of R-n, n >= 2, is an open, bounded and convex set such that BR1 (sic) Omega(0), then the first Steklov-Dirichlet eigenvalue sigma(1)(Omega) has a maximum when R-1 and the measure of Omega are fixed. Moreover, if Omega(0) is contained in a suitable ball, we prove that the spherical shell is the maximum.}, author = {Gavitone, Nunzia and Paoli, Gloria and Piscitelli, Gianpaolo and Sannipoli, Rossano}, doi = {10.2140/pjm.2022.320.241}, faupublication = {yes}, journal = {Pacific Journal of Mathematics}, note = {CRIS-Team WoS Importer:2023-03-03}, pages = {241-259}, peerreviewed = {Yes}, title = {{AN} {ISOPERIMETRIC} {INEQUALITY} {FOR} {THE} {FIRST} {STEKLOV}-{DIRICHLET} {LAPLACIAN} {EIGENVALUE} {OF} {CONVEX} {SETS} {WITH} {A} {SPHERICAL} {HOLE}}, volume = {320}, year = {2022} } @article{faucris.307431895, abstract = {The optimal control of thermally convective flows is usually modeled by an optimization problem with constraints of Boussinesq equations that consist of the Navier-Stokes equation and an advection-diffusion equation. This optimal control problem is challenging from both theoretical analysis and algorithmic design perspectives. For example, the nonlinearity and coupling of fluid flows and energy transports prevent direct applications of gradient type algorithms in practice. In this paper, we propose an efficient numerical method to solve this problem based on the operator splitting and optimization techniques. In particular, we employ the Marchuk-Yanenko method leveraged by the $L^2-$projection for the time discretization of the Boussinesq equations so that the Boussinesq equations are decomposed into some easier linear equations without any difficulty in deriving the corresponding adjoint system. Consequently, at each iteration, four easy linear advection-diffusion equations and two degenerated Stokes equations at each time step are needed to be solved for computing a gradient. Then, we apply the Bercovier-Pironneau finite element method for space discretization, and design a BFGS type algorithm for solving the fully discretized optimal control problem. We look into the structure of the problem, and design a meticulous strategy to seek step sizes for the BFGS efficiently. Efficiency of the numerical approach is promisingly validated by the results of some preliminary numerical experiment}, author = {Song, Yongcun and Yuan, Xiaomig and Yue, Hangrui}, faupublication = {yes}, journal = {Journal of Computational Physics}, peerreviewed = {Yes}, title = {{A} numerical approach to the optimal control of thermally convective flows}, url = {https://arxiv.org/abs/2211.15302}, year = {2023} } @incollection{faucris.235595523, abstract = {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}, author = {Ruiz-Balet, Domènec and Zuazua, Enrique}, booktitle = {Applied Wave Mathematics II}, doi = {10.1007/978-3-030-29951-4}, editor = {Arkadi Berezovski, Tarmo Soomere}, faupublication = {yes}, isbn = {978-3-030-29950-7}, pages = {343-363}, peerreviewed = {unknown}, publisher = {Springer}, series = {Mathematics of Planet Earth}, title = {{A} {Parabolic} {Approach} to the {Control} of {Opinion} {Spreading}}, volume = {6}, year = {2019} } @unpublished{faucris.310887833, abstract = {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.

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.

We consider systems that are governed by linear time-discrete dynamics with an initial condition and a terminal condition for the expected values. We study optimal control problems where in the objective function a term of tracking type for the expected values and a control cost appear. In addition, the feasible states have to satisfy a conservative probabilistic constraint that requires that the probability that the trajectories remain in a given set F is greater than or equal to a given lower bound. An application are optimal control problems related to storage management systems with uncertain in- and output. We give sufficient conditions that imply that the optimal expected trajectories remain close to a certain state that can be characterized as the solution of an optimal control problem without prescribed initial- and terminal condition. In this way we contribute to the study of the turnpike phenomenon that is well-known in mathematical economics and make a step towards the extension of the turnpike theory to problems with probabilistic constraints.

0$. We show that as long as the $H^1$-norm of the function that describes the noise in the customer's behavior decays exponentially with a rate that is sufficiently large, the velocity of the gas can be stabilized exponentially fast in the sense that a suitably chosen Lyapunov function decays exponentially. For the exponential stability it is sufficient that the feedback parameter $k$ is sufficiently large and the stationary state to which the system is stabilized is sufficiently small. The stability result is local, that is, it holds for initial states that are sufficiently close to the stationary state. This result is an example for the exponential boundary feedback stabilization of a quasi-linear hyperbolic system with uncertain boundary data. The analysis is based upon the choice of a suitably Lyapunov function. The decay of this Lyapunov function implies that also the $L^2$-norm of the difference of the system state and the stationary state decays exponentially.

Read More: https://epubs.siam.org/doi/10.1137/16M1090156

= 5 is odd: therein, we show that the boundary of the diagram is given by the graphs of two continuous and strictly increasing functions. An explicit formula for the lower one and a numerical method to obtain the upper one is provided. At last, some applications of the results are presented.}, author = {Ftouhi, Ilias}, doi = {10.1142/S0219199722500547}, faupublication = {yes}, journal = {Communications in Contemporary Mathematics}, note = {CRIS-Team WoS Importer:2022-09-30}, peerreviewed = {Yes}, title = {{Complete} systems of inequalities relating the perimeter, the area and the {Cheeger} constant of planar domains}, year = {2022} } @misc{faucris.236486417, abstract = {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.

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 \leq u \leq 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.

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.

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},
author = {Biccari, Umberto and Warma, Mahamadi and Zuazua Iriondo, Enrique},
doi = {10.1016/bs.hna.2021.12.001},
faupublication = {yes},
journal = {Handbook of Numerical Analysis},
keywords = {Fractional Laplacian; Fractional diffusion equation; Finite Element Method; Interior Control; Exterior Control; Simultaneous Control; Numerical approximation},
month = {Jan},
peerreviewed = {unknown},
title = {{Control} and {Numerical} approximation of {Fractional} {Diffusion} {Equations}},
url = {https://dcn.nat.fau.eu/wp-content/uploads/BiccariWarmaZuazua{\_}fractionalControl.pdf},
year = {2022}
}
@article{faucris.260140114,
abstract = {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.},
author = {Maity, Debayan and Tucsnak, Marius and Zuazua, Enrique},
faupublication = {yes},
journal = {Control and Cybernetics},
keywords = {Admissible control operator; Age structure; Infinite dimensional linear system; Null controllability; Population dynamics},
note = {CRIS-Team Scopus Importer:2021-06-15},
pages = {231-260},
peerreviewed = {Yes},
title = {{Controllability} of a class of infinite dimensional systems with age structure},
volume = {48},
year = {2019}
}
@misc{faucris.236483424,
abstract = {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.

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.},
author = {Biccari, Umberto and Warma, Mahamadi and Zuazua, Enrique},
doi = {10.3934/cpaa.2020086},
faupublication = {yes},
journal = {Communications on Pure and Applied Analysis},
note = {CRIS-Team WoS Importer:2020-01-31},
pages = {1949-1978},
peerreviewed = {Yes},
title = {{Controllability} of the one-dimensional fractional heat equation under positivity constraints},
volume = {19},
year = {2020}
}
@article{faucris.236489408,
abstract = {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.

},
author = {Ruiz-Balet, Domènec and Zuazua, Enrique},
doi = {10.3934/mcrf.2022032},
faupublication = {yes},
journal = {Mathematical Control and Related Fields},
keywords = {Reaction-diffusion, Control, Steady-states, Constraints, Mathematical Biology},
pages = {955-1038},
peerreviewed = {Yes},
title = {{Control} of certain parabolic models from biology and social sciences},
volume = {12},
year = {2022}
}
@unpublished{faucris.319275701,
abstract = {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.

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},
author = {Ruiz-Balet, Domènec and Zuazua Iriondo, Enrique},
doi = {10.3934/mcrf.2022032},
faupublication = {yes},
keywords = {Reaction-diffusion; control; paths of steady-states; constraints; phase plane; traveling waves; comparison principle; Mathematical Biology},
note = {https://cris.fau.de/converis/publicweb/Publication/276859119},
peerreviewed = {automatic},
title = {{Control} of reaction-diffusion models in biology and social sciences},
url = {https://dcn.nat.fau.eu/wp-content/uploads/NotesSichuan-DRuizBalet{\_}EZuazua.pdf},
year = {2024}
}
@misc{faucris.236485353,
abstract = {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.

n→R^{m} satisfying given equality or inequality constraints. Each constraint may be imposed over Ω or its boundary, either pointwise or in an integral sense. These global minimizations are generally non-convex and intractable. We formulate a particular convex maximization, here called the pointwise dual relaxation (PDR), whose supremum is a lower bound on the infimum of the
original problem. The PDR can be derived by dualizing and relaxing the original problem; its constraints are pointwise equalities or inequalities over finite-dimensional sets, rather than over infinite-dimensional function spaces. When the original minimization can be specified by polynomial functions of (x,u,∇u), the PDR can be further relaxed by replacing pointwise inequalities with polynomial sum-of-squares (SOS) conditions. The resulting SOS program is computationally tractable when the dimensions m,n and number of constraints are not too large. The framework presented here generalizes an approach of Valmorbida, Ahmadi, and Papachristodoulou (IEEE Trans. Automat. Contr., 61:1649--1654, 2016). We prove that the optimal lower bound given by the PDR is sharp for several classes of problems, whose special cases include leading eigenvalues of Sturm-Liouville problems and optimal constants of Poincaré inequalities. For these same classes, we prove that SOS relaxations of the PDR converge to the sharp lower bound as polynomial degrees are increased. Convergence of SOS computations in practice is illustrated for several example},
author = {Fantuzzi, Giovanni and Chernyavsky, Alexandr and Goluskin, David and Bramburger, Jason},
doi = {10.1137/21m1455127},
faupublication = {no},
journal = {SIAM Journal on Optimization},
peerreviewed = {Yes},
title = {{Convex} {Relaxations} of {Integral} {Variational} {Problems}: {Pointwise} {Dual} {Relaxation} and {Sum}-of-{Squares} {Optimization}},
year = {2023}
}
@article{faucris.298892075,
abstract = {For s?(0,1),N>2s, and a bounded open set (O)?R(N)with C(2)boundary, we study the fractional Brezis-Nirenberg type minimization problem of finding},
author = {De Nitti, Nicola and Konig, Tobias},
doi = {10.1007/s00526-023-02446-1},
faupublication = {yes},
journal = {Calculus of Variations and Partial Differential Equations},
note = {CRIS-Team WoS Importer:2023-05-05},
peerreviewed = {Yes},
title = {{Critical} functions and blow-up asymptotics for the fractional {Brezis}-{Nirenberg} problem in low dimension},
volume = {62},
year = {2023}
}
@article{faucris.284133628,
abstract = {We study the regularity of the flow X(t, y) , which represents (in the sense of Smirnov or as regular Lagrangian flow of Ambrosio) a solution ρ∈ L^{∞}(R^{d}^{+}^{1}) of the continuity equation ∂tρ+div(ρb)=0,with b∈Lt1BVx. We prove that X is differentiable in measure in the sense of Ambrosio–Malý, that is X(t,y+rz)-X(t,y)r→r→0W(t,y)zin measure,where the derivative W(t, y) is a BV function satisfying the ODE ddtW(t,y)=(Db)y(dt)J(t-,y)W(t-,y),where (Db) y(d t) is the disintegration of the measure ∫Db(t,·)dt with respect to the partition given by the trajectories X(t, y) and the Jacobian J(t, y) solves ddtJ(t,y)=(divb)y(dt)=Tr(Db)y(dt).The proof of this regularity result is based on the theory of Lagrangian representations and proper sets introduced by Bianchini and Bonicatto in [16], on the construction of explicit approximate tubular neighborhoods of trajectories, and on estimates that take into account the local structure of the derivative of a BV vector field.},
author = {Bianchini, Stefano and De Nitti, Nicola},
doi = {10.1007/s00205-022-01820-1},
faupublication = {yes},
journal = {Archive for Rational Mechanics and Analysis},
note = {CRIS-Team Scopus Importer:2022-10-28},
peerreviewed = {Yes},
title = {{Differentiability} in {Measure} of the {Flow} {Associated} with a {Nearly} {Incompressible} {BV} {Vector} {Field}},
year = {2022}
}
@article{faucris.283175916,
abstract = {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.},
author = {Esteve-Yague, Carlos and Zuazua, Enrique},
doi = {10.1137/22M1469353},
faupublication = {yes},
journal = {SIAM Journal on Mathematical Analysis},
month = {Jan},
note = {CRIS-Team WoS Importer:2022-10-14},
pages = {5388-5423},
peerreviewed = {Yes},
title = {{Differentiability} {With} {Respect} {To} {The} {Initial} {Condition} {For} {Hamilton}-jacobi {Equations}},
volume = {54},
year = {2022}
}
@article{faucris.265423753,
abstract = {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.

}, author = {Esteve-Yague, Carlos and Zuazua Iriondo, Enrique}, doi = {10.1137/22M1469353}, faupublication = {yes}, journal = {SIAM Journal on Mathematical Analysis}, keywords = {Hamilton-Jacobi equation; Gâteaux derivatives; inverse design problem; transport equation; duality solutions}, pages = {5388-5423}, peerreviewed = {Yes}, title = {{Differentiability} with respect to the initial condition for {Hamilton}-{Jacobi} equations}, volume = {54}, year = {2022} } @article{faucris.289674511, abstract = {We study the diffusive relaxation limit of the Jin-Xin system toward viscous conservation laws in the multi-dimensional setting. For initial data being small perturbations of a constant state in suitable homogeneous Besov norms, we prove the global well-posedness of strong solutions satisfying uniform estimates with respect to the relaxation parameter. Then, we justify the strong relaxation limit and exhibit an explicit convergence rate of the process. Our proof is based on an adaptation of the techniques developed in [12,13] to be able to deal with additional low-order nonlinear terms.}, author = {Crin-Barat, Timothée and Shou, Ling Yun}, doi = {10.1016/j.jde.2023.02.015}, faupublication = {yes}, journal = {Journal of Differential Equations}, keywords = {Besov spaces; Hyperbolic approximation; Jin-Xin approximation; Relaxation limit}, note = {CRIS-Team Scopus Importer:2023-02-24}, pages = {302-331}, peerreviewed = {Yes}, title = {{Diffusive} relaxation limit of the multi-dimensional {Jin}-{Xin} system}, volume = {357}, year = {2023} } @article{faucris.319954760, author = {Djida, Jean-Daniel}, doi = {10.1090/proc/16702}, faupublication = {yes}, journal = {Proceedings of the American Mathematical Society}, peerreviewed = {Yes}, title = {{Domination} of semigroups generated by regular forms}, url = {https://www.ams.org/journals/proc/0000-000-00/S0002-9939-2024-16702-2/}, year = {2024} } @article{faucris.255832569, abstract = {We consider the problem of steering a finite string to the zero state in finite time from a given initial state by controlling the state at one boundary point while the other boundary point moves. As a possible application we have in mind the optimal control of a mining elevator, where the length of the string changes during the transportation process. During the transportation pro-cess, oscillations of the elevator-cable can occur that can be damped in this way. We present an exact controllability result for Dirichlet boundary control at the fixed end of the string that states that there exist exact controls for which the oscillations vanish after finite time. For the result we assume that the movements are Lipschitz continuous with a Lipschitz constant, whose absolute value is smaller than the wave speed. In the result, we present the minimal time, for which exact controllability holds, this time depending on the movement of the boundary point. Our results are based upon travelling wave so-lutions. We present a representation of the set of successful controls that steer the system to rest after finite time as the solution set of two point-wise equalities. This allows for a transformation of the optimal control problem to a form where no partial differential equation appears. This representation enables interesting insights into the structure of the successful controls. For example, exact bang-bang controls can only exist if the initial state is a simple function and the initial velocity is zero.}, author = {Gugat, Martin}, faupublication = {yes}, journal = {Control and Cybernetics}, keywords = {Analytic solution; Exact controllability; Mining elevator; Moving boundaries; Optimal boundary control; Optimal control of pdes; Pde constrained optimization; Wave equation}, note = {CRIS-Team Scopus Importer:2021-04-20}, pages = {69-87}, peerreviewed = {Yes}, title = {{Exact} controllability of a string to rest with a moving boundary}, volume = {48}, year = {2019} } @inproceedings{faucris.310750644, abstract = {In Higher Engineering Mathematics (HEM) several concepts are hard to sketch at the blackboard. Starting from stationary points at graphs of function with two unknowns or points with certain properties of the Hessian up to vector field or how coordinate transformations work are difficult the sketch.

JSXGraph now supports the visualization of 3D interactive diagrams as a brand new feature.

The talk will show some applications demonstrating the world of 3D JSXGraph and its use within the STACK question type.

The focus will be set on

- coordinate transformation:
- polar coordinates,
- spherical coordinates,
- general coordinate transformation,

- calculus of function with two variable,
- visualization of vector fields.

- 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}, author = {Liard, Thibault and Zuazua, Enrique}, faupublication = {yes}, keywords = {Inverse problems; Conservation Laws; Entropy solutions; Backward-Forward approach; Optimal Control Problem; Wave-front tracking algorithm.}, peerreviewed = {automatic}, title = {{Inverse} design for the one-dimensional {Burgers} equation}, year = {2019} } @unpublished{faucris.310644407, author = {Crin-Barat, Timothée and Zuazua Iriondo, Enrique and et al.}, author_hint = {Crin-Barat T, Shou L-Y, Zuazua E}, doi = {doi.org/10.48550/arXiv.2308.08280}, faupublication = {yes}, note = {https://cris.fau.de/converis/publicweb/Publication/310644407}, peerreviewed = {automatic}, support_note = {Author relations incomplete. You may find additional data in field 'author{\_}hint'}, title = {{Large} time asymptotics for partially dissipative hyperbolic systems without {Fourier} analysis: application to the nonlinearly damped p-system}, year = {2024} } @article{faucris.265989500, abstract = {The Goldstein-Taylor equations can be thought of as a simplified version of a BGK system, where the velocity variable is constricted to a discrete set of values. It is intimately related to turbulent fluid motion and the telegrapher’s equation. A detailed understanding of the large time behaviour of the solutions to these equations has been mostly achieved in the case where the relaxation function, measuring the intensity of the relaxation towards equally distributed velocity densities, is constant. The goal of the presented work is to provide a general method to tackle the question of convergence to equilibrium when the relaxation function is not constant, and to do so as quantitatively as possible. In contrast to the usual modal decomposition of the equations, which is natural when the relaxation function is constant, we define a new Lyapunov functional of pseudodifferential nature, one that is motivated by the modal analysis in the constant case, that is able to deal with full spatial dependency of the relaxation function. The approach we develop is robust enough that one can apply it to multi-velocity Goldstein-Taylor models, and achieve explicit rates of convergence. The convergence rate we find, however, is not optimal, as we show by comparing our result to those found in [8].}, author = {Arnold, Anton and Einav, Amit and Signorello, Beatrice and Wöhrer, Tobias}, doi = {10.1007/s10955-021-02702-8}, faupublication = {no}, journal = {Journal of Statistical Physics}, keywords = {BGK equation; Exponential decay; Hypocoercivity; Large time behaviour; Lyapunov functional}, note = {Created from Fastlane, Scopus look-up}, peerreviewed = {Yes}, title = {{Large} {Time} {Convergence} of the {Non}-homogeneous {Goldstein}-{Taylor} {Equation}}, volume = {182}, year = {2021} } @incollection{faucris.270704667, abstract = {We present a positive and negative stabilization results for a semilinear model of gas flow in pipelines.

For feedback boundary conditions, we obtain un-conditional stabilization in the absence and conditional instability in the presence of the source term.

We also obtain unconditional instability for the corresponding quasilinear model given by the isothermal Euler equations.

We present a positive and negative stabilization results for a semilinearmodel of gas flow in pipelines. For feedback boundary conditions, we obtain un-conditional stabilization in the absence and conditional instability in the presenceof the source term. We also obtain unconditional instability for the correspondingquasilinearmodelgivenbytheisothermalEulerequationsWe present a positive and negative stabilization results for a semilinearmodel of gas flow in pipelines. For feedback boundary conditions, we obtain un-conditional stabilization in the absence and conditional instability in the presenceof the source term. We also obtain unconditional instability for the correspondingquasilinearmodelgivenbytheisothermalEulerequationsWe present a positive and negative stabilization results for a semilinearmodel of gas flow in pipelines. For feedback boundary conditions, we obtain un-conditional stabilization in the absence and conditional instability in the presenceof the source term. We also obtain unconditional instability for the correspondingquasilinearmodelgivenbytheisothermalEulerequatio},
author = {Gugat, Martin and Herty, Michael},
booktitle = {Optimization and Control for Partial Differential Equations},
doi = {10.1515/9783110695984-003},
editor = {Roland Herzog, Matthias Heinkenschloss, Dante Kalise, Georg Stadler und Emmanuel Trélat},
faupublication = {yes},
isbn = {9783110695984},
keywords = {stabilization; hyperbolic partial differential equations; feedback law;},
pages = {59-71},
peerreviewed = {Yes},
publisher = {De Gruyter},
series = {Radon Series on Computational and Applied Mathematics},
title = {{Limits} of stabilizability for a semilinear model for gas pipeline flow},
volume = {29},
year = {2022}
}
@article{faucris.316303753,
abstract = {This paper is devoted to the discussion of the exponential stability of a networked hyperbolic system with a circle. Our analysis extends an example by Bastin and Coron about the limits of boundary stabilizability of hyperbolic systems to the case of a networked system that is defined on a graph which contains a cycle.

By spectral analysis, we prove that the system is stabilizable while the length of the arcs is sufficiently small. However, if the length of the arcs is too large, the system is not stabilizable. Our results are robust with respect to small perturbations of the arc lengths.

Complementing our analysis, we provide numerical simulations that illustrate our findings.

},
author = {Coclite, Giuseppe Maria and De Nitti, Nicola and Keimer, Alexander and Pflug, Lukas and Zuazua Iriondo, Enrique},
doi = {10.1088/1361-6544/acf01d},
faupublication = {yes},
journal = {Nonlinearity},
keywords = {Nonlocal conservation laws; nonlocal flux; Burgers’ equation; approximation of local conservation laws; N-waves; source-type solutions; entropy solutions},
peerreviewed = {Yes},
title = {{Long}-time convergence of a nonlocal {Burgers}' equation towards the local {N}-wave},
url = {https://iopscience.iop.org/article/10.1088/1361-6544/acf01d},
year = {2023}
}
@inproceedings{faucris.289827009,
abstract = {In the transition to renewable energy sources, hydrogen will potentially play an important role for energy storage. The efficient transport of this gas is possible via pipelines. An understanding of the possibilities to control the gas flow in pipelines is one of the main building blocks towards the optimal use of gas. For the operation of gas transport networks it is important to take into account the randomness of the consumers' demand, where often information on the probability distribution is available. Hence in an efficient optimal control model the corresponding probability should be included and the optimal control should be such that the state that is generated by the optimal control satisfies given state constraints with large probability. We comment on the modelling of gas pipeline flow and the problems of optimal nodal control with random demand, where the aim of the optimization is to determine controls that generate states that satisfy given pressure bounds with large probability. We include the H^2 norm of the control as control cost, since this avoids large pressure fluctuations which are harmful in the transport of hydrogen since they can cause embrittlement of the pipeline meta},
author = {Schuster, Michael and Gugat, Martin},
booktitle = {Extended Abstracts presented at the 25th International Symposium on Mathematical Theory of Networks and Systems MTNS 2022},
date = {2022-09-12/2022-09-16},
doi = {10.15495/EPub{\_}UBT{\_}00006809},
editor = {Baumann, Michael Heinrich; Grüne, Lars; Jacob, Birgit; Worthmann, Karl},
faupublication = {yes},
keywords = {gas pipeline flow, nodal control, boundary control, optimal control, hyperbolic differential equation, random demand, state constraints, pressure bound, classical solutions},
peerreviewed = {unknown},
title = {{Max}-p {Optimal} {Boundary} {Control} of {Gas} {Flow}},
venue = {Bayreuth},
year = {2022}
}
@incollection{faucris.268203529,
abstract = {In this chapter we survey recent progress on mathematical results on gas flow in pipe networks with a special focus on questions of control and stabilization. We briefly present the modeling of gas flow and coupling conditions for flow through vertices of a network. Our main focus is on gas models for spatially one-dimensional flow governed by hyperbolic balance laws. We survey results on classical solutions as well as weak solutions. We present results on well-posedness, controllability, feedback stabilization, the inclusion of uncertainty in the models and numerical methods.

0, whose levels blow-up as T→0+. Moreover, this method leads to efficient numerical algorithms for computing multilevel controls.

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.

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.

The flow is governed by a quasilinear hyperbolic model.

Since in the operation of the gas networks regular solutions without shocks are desirable, we imposeappropriate state and control constraint in order to guarantee that a classical solution is generated.

Due to a $W^{2,\infty}$-regularization term in the objective function, we can show the existence of an optimal control.

Moreover, we give conditions that guarantee that the control becomes constant a the end of the control time interval if the weight of the regularization term is sufficiently larg},
author = {Gugat, Martin and Sokolowski, Jan},
faupublication = {yes},
journal = {Pure and Applied Functional Analysis},
keywords = {gas pipeline flow; optimal control; regular solutions;},
pages = {1699-1715},
peerreviewed = {Yes},
title = {{On} problems of dynamic optimal nodal control for gas networks},
url = {http://yokohamapublishers.jp/online2/oppafa/vol7/p1699.html},
volume = {5},
year = {2022}
}
@article{faucris.308878212,
author = {Ftouhi, Ilias},
doi = {10.1016/j.jmaa.2021.125443},
faupublication = {no},
journal = {Journal of Mathematical Analysis and Applications},
peerreviewed = {Yes},
title = {{On} the {Cheeger} inequality for convex sets},
url = {https://hal.science/hal-03006015/document},
year = {2021}
}
@article{faucris.287354212,
abstract = {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.

give rise to a turnpike structure in the sense that for natural numbers n if T is sufficiently large, the contribution of the objective function from subintervals of [0, T] of the form

[t−t/2n,t+(T−t)/2n]

is of the order 1/min{tn,(T−t)n}.

We also show that a similar result holds for ϵ-optimal solutions of the optimal control problems if ϵ>0 is chosen sufficiently small. At the end of the paper we present both systems that are governed by ordinary differential equations and systems governed by partial differential equations where the results can be applied.

OPTIMAL BOUNDARY CONTROL OF THE WAVE EQUATION:

THE FINITE-TIME TURNPIKE PHENOMENON

MARTIN GUGAT

Communicated by Nicolae Cˆındea

It is well-known that vibrating strings can be steered to a position of rest in

finite time by suitably defined boundary control functions, if the time horizon is

sufficiently long. In optimal control problems, the desired terminal state is often

enforced by terminal conditions, that add an additional difficulty to the optimal

control problem. In this paper we present an optimal control problem for the

wave equation with a time-dependent weight in the objective function such that

for a sufficiently long time horizon, the optimal state reaches a position of rest in

finite time without prescribing a terminal constraint. This situation can be seen

as a realization of the finite-time turnpike phenomenon that has been studied

recently.

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},
author = {Gugat, Martin and Zuazua Iriondo, Enrique and Schuster, Michael},
booktitle = {Extended Abstracts of the 65th Joint Conference on Automatic Control},
date = {2022-11-12/2022-11-13},
faupublication = {yes},
keywords = {Turnpike phenomenon, hyperbolic differential equation, optimal control, boundary control, random boundary data},
peerreviewed = {unknown},
title = {{Optimal} {Control} for {ODEs} and {PDEs}: {The} {Turnpike} {Phenomenon}},
venue = {Utsunomiya},
year = {2022}
}
@article{faucris.265993865,
abstract = {The paper is concerned with an optimal control problem governed by a state equation in form of a generalized abstract operator differential equation involving a maximal monotone operator. The state equation is uniquely solvable, but the associated solution operator is in general not Gâteaux-differentiable. In order to derive optimality conditions, we therefore regularize the state equation and its solution operator, respectively, by means of a (smoothed) Yosida approximation. We show convergence of global minimizers for regularization parameter tending to zero and derive necessary and sufficient optimality conditions for the regularized problems. The paper ends with an application of the abstract theory to optimal control of homogenized quasi-static elastoplasticity.

},
author = {Meinlschmidt, Hannes and Meyer, Christian and Walter, Stephan},
doi = {10.46298/jnsao-2020-5800},
faupublication = {yes},
journal = {Journal of Nonsmooth Analysis and Optimization},
keywords = {Optimal control; variational inequality; homogenized pasticity},
peerreviewed = {Yes},
title = {{Optimal} control of an abstract evolution variational inequality with application in homogenized plasticity},
url = {https://jnsao.episciences.org/6467},
volume = {1},
year = {2020}
}
@article{faucris.313066707,
abstract = {We show a turnpike result for problems of optimal control with possibly nonlinear systems as well as pointwise-in-time state and control constraints. The objective functional is of integral type and contains a tracking term which penalizes the distance to a desired steady state. In the optimal control problem, only the initial state is prescribed. We assume that a cheap control condition holds that yields a bound for the optimal value of our optimal control problem in terms of the initial data. We show that the solutions to the optimal control problems on the time intervals [0, *T*] have a turnpike structure in the following sense: For large T the contribution to the objective functional that comes from the subinterval [*T*/2, *T*], i.e., from the second half of the time interval [0, T], is at most of the order 1/*T*. More generally, the result holds for subintervals of the form [*r T,T*], where *r* ∈ (0, 1/2) is a real number. Using this result inductively implies that the decay of the integral on such a subinterval in the objective function is faster than the reciprocal value of a power series in *T* with positive coefficients. Accordingly, the contribution to the objective value from the final part of the time interval decays rapidly with a growing time horizon. At the end of the paper we present examples for optimal control problems where our results are applicable.

inf{dH(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}, author = {Ftouhi, Ilias and Zuazua Iriondo, Enrique}, doi = {10.1007/s12220-023-01301-1}, faupublication = {yes}, journal = {Journal of Geometric Analysis}, keywords = {Shape optimization; Convex geometry; sensor design}, month = {Jan}, peerreviewed = {Yes}, title = {{Optimal} design of sensors via geometric criteria}, url = {https://link.springer.com/article/10.1007/s12220-023-01301-1}, volume = {33}, year = {2023} } @article{faucris.241650251, abstract = {

Abstract | References | PDF (618 KB) | ||

In optimal control problems, often initial data are required that are not known exactly in practice. In order to take into account this uncertainty, we consider optimal control problems for a system with an uncertain initial state. A finite terminal time is given. On account of the uncertainty of the initial state, it is not possible to prescribe an exact terminal state. Instead, we are looking for a control that steers the system into a given neighborhood of the desired terminal state with sufficiently high probability. This neighborhood is described in terms of an inequality for the terminal energy. The probabilistic constraint in the considered optimal control problem leads to optimal controls that are robust against the inevitable uncertainties of the initial state. Numerical examples with optimal Neumann control of the wave equation are presented. |

The strict positivity of the wave velocity allows for the dynamics in the unconstrained region R−" role="presentation" style="display: inline-block; line-height: 0; font-size: 19.04px; overflow-wrap: normal; word-spacing: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: initial; padding-top: 1px; padding-bottom: 1px; position: relative;">R−�− to be fully determined by the restriction of the initial data to R−" role="presentation" style="display: inline-block; line-height: 0; font-size: 19.04px; overflow-wrap: normal; word-spacing: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: initial; padding-top: 1px; padding-bottom: 1px; position: relative;">R−�−.

On the other hand, the solution in the constrained region is dictated by the assumption that the total mass of the initial datum is conserved along the evolution. We formulate the transmission condition at the interface {x=0}" role="presentation" style="display: inline-block; line-height: 0; font-size: 19.04px; overflow-wrap: normal; word-spacing: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: initial; padding-top: 1px; padding-bottom: 1px; position: relative;">{x=0}{�=0} in such a way that the boundary datum for the initial boundary value problem posed on R+" role="presentation" style="display: inline-block; line-height: 0; font-size: 19.04px; overflow-wrap: normal; word-spacing: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: initial; padding-top: 1px; padding-bottom: 1px; position: relative;">R+�+ is given by the largest incoming flux that is admissible under the constraint, while the exceeding mass is accumulated in a " role="presentation" style="display: inline-block; line-height: 0; font-size: 19.04px; overflow-wrap: normal; word-spacing: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: initial; padding-top: 1px; padding-bottom: 1px; position: relative;">buffer'' (as an atomic measure concentrated at the interface},
author = {De Nitti, Nicola and Serre, Denis and Zuazua Iriondo, Enrique},
faupublication = {yes},
note = {https://cris.fau.de/converis/publicweb/Publication/319465719},
peerreviewed = {automatic},
title = {{Pointwise} constraints for scalar conservation laws with positive wave velocity},
url = {http://cvgmt.sns.it/paper/6472/},
year = {2024}
}
@article{faucris.294849924,
abstract = {In this paper, we study a singular limit problem for a compressible one-velocity bifluid system. More precisely, we show that solutions of the Kapila system generated by initial data close to equilibrium are obtained in the pressure-relaxation limit from solutions of the Baer-Nunziato (BN) system. The convergence rate of this process is a consequence of our stability result. Besides the fact that the quasilinear part of the (BN) system cannot be written in conservative form, its natural associated entropy is only positive semi-definite such that it is not clear if the entropic variables can be used in the present case. Using an ad-hoc change of variables we obtain a reformulation of the (BN) system which couples, via low-order terms, an undamped mode and a non-symmetric partially dissipative hyperbolic system satisfying the Shizuta-Kawashima stability condition.},
author = {Burtea, Cosmin and Crin-Barat, Timothée and Tan, Jin},
doi = {10.1142/S0218202523500161},
faupublication = {yes},
journal = {Mathematical Models & Methods in Applied Sciences},
keywords = {Baer-Nunziato; Kapila system; non-conservative quasilinear systems; one-velocity bifluid system; pressure-relaxation limit},
note = {CRIS-Team Scopus Importer:2023-03-31},
peerreviewed = {Yes},
title = {{Pressure}-relaxation limit for a one-velocity {Baer}-{Nunziato} model to a {Kapila} model},
year = {2023}
}
@article{faucris.256693794,
abstract = {Uncertainty often plays an important role in dynamic flow problems. In this paper, we consider both, a stationary and a dynamic flow model with uncertain boundary data on networks. We introduce two different ways how to compute the probability for random boundary data to be feasible, discussing their advantages and disadvantages. In this context, feasible means, that the flow corresponding to the random boundary data meets some box constraints at the network junctions. The first method is the spheric radial decomposition and the second method is a kernel density estimation. In both settings, we consider certain optimization problems and we compute derivatives of the probabilistic constraint using the kernel density estimator. Moreover, we derive necessary optimality conditions for an approximated problem for the stationary and the dynamic case. Throughout the paper, we use numerical examples to illustrate our results by comparing them with a classical Monte Carlo approach to compute the desired probability.

−2, where R is a nondimensional `flux' Rayleigh number quantifying the strength of internal heating and c=216. Then, \overline{⟨wT⟩}=0 corresponds to vertical heat transport by conduction alone, while \overline{⟨wT⟩}>0 represents the enhancement of vertical heat transport upwards due to convective motion. If, instead, the lower boundary is a thermal insulator, then we obtain \overline{⟨wT⟩}≤1/2−cR^{−4}, with c≈0.0107. This result implies that the Nusselt number Nu, defined as the ratio of the total-to-conductive heat transport, satisfies Nu≲R4. Both bounds are obtained by combining the background method with a minimum principle for the fluid's temperature and with Hardy-Rellich inequalities to exploit the link between the vertical velocity and temperature. In both cases, power-law dependence on R improves the previously best-known bounds, which, although valid at both infinite and finite Prandtl numbers, approach the uniform bound exponentially with R.},
author = {Arslan, Ali and Fantuzzi, Giovanni and John, Craske and Wynn, Andrew},
doi = {10.1063/5.0098250},
faupublication = {no},
journal = {Journal of Mathematical Physics},
keywords = {Heat transfer, Mathematical optimization, Auxiliary functions, Calculus of variations, Fluid mechanics, Natural convection},
peerreviewed = {Yes},
title = {{Rigorous} scaling laws for internally heated convection at infinite {Prandtl} number},
volume = {64},
year = {2023}
}
@article{faucris.239808884,
abstract = {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.},
author = {Lance, Gontran and Trélat, Emmanuel and Zuazua, Enrique},
doi = {10.1016/j.sysconle.2020.104733},
faupublication = {yes},
journal = {Systems & Control Letters},
keywords = {Direct methods; Optimal shape design; Parabolic equation; Strict dissipativity; Turnpike},
note = {CRIS-Team Scopus Importer:2020-06-30},
peerreviewed = {Yes},
title = {{Shape} turnpike for linear parabolic {PDE} models},
volume = {142},
year = {2020}
}
@article{faucris.276875174,
abstract = {We establish sharp criteria for the instantaneous propagation of free boundaries in solutions to the thin-film equation. The criteria are formulated in terms of the initial distribution of mass (as opposed to previous almost-optimal results), reflecting the fact that mass is a locally conserved quantity for the thin-film equation. In the regime of weak slippage, our criteria are at the same time necessary and sufficient. The proof of our upper bounds on free boundary propagation is based on a strategy of "propagation of degeneracy" down to arbitrarily small spatial scales: We combine estimates on the local mass and estimates on energies to show that "degeneracy" on a certain space-time cylinder entails "degeneracy" on a spatially smaller space-time cylinder with the same time horizon. The derivation of our lower bounds on free boundary propagation is based on a combination of a monotone quantity and almost optimal estimates established previously by the second author with a new estimate connecting motion of mass to entropy production.},
author = {De Nitti, Nicola and Fischer, Julian},
doi = {10.1080/03605302.2022.2056702},
faupublication = {yes},
journal = {Communications in Partial Differential Equations},
note = {CRIS-Team WoS Importer:2022-06-17},
peerreviewed = {Yes},
title = {{Sharp} criteria for the waiting time phenomenon in solutions to the thin-film equation},
year = {2022}
}
@article{faucris.265989248,
abstract = {We review the Lyapunov functional method for linear ODEs and give an explicit construction of such functionals that yields sharp decay estimates, including an extension to defective ODE systems. As an application, we consider three evolution equations, namely the linear convection-diffusion equation, the two velocity BGK model and the Fokker–Planck equation. Adding an uncertainty parameter to the equations and analyzing its linear sensitivity leads to defective ODE systems. By applying the Lyapunov functional construction, we prove sharp long time behavior of order (1+t^{M})e^{−μt}, where M is the defect and μ is the spectral gap of the system. The appearance of the uncertainty parameter in the three applications makes it important to have decay estimates that are uniform in the non-defective limit.},
author = {Arnold, Anton and Jin, Shi and Wöhrer, Tobias},
doi = {10.1016/j.jde.2019.08.047},
faupublication = {no},
journal = {Journal of Differential Equations},
keywords = {Defective ODEs; Kinetic equations; Long time behavior; Lyapunov functionals; Sensitivity analysis; Uncertainty quantification},
month = {Jan},
note = {Created from Fastlane, Scopus look-up},
pages = {1156-1204},
peerreviewed = {Yes},
title = {{Sharp} decay estimates in local sensitivity analysis for evolution equations with uncertainties: {From} {ODEs} to linear kinetic equations},
volume = {268},
year = {2020}
}
@incollection{faucris.265887115,
author = {Wöhrer, Tobias and Dolbeault, Jean and Arnold, Anton and Schmeiser, Christian},
booktitle = {Recent Advances in Kinetic Equations and Applications},
doi = {10.1007/978-3-030-82946-9{\_}1},
faupublication = {yes},
peerreviewed = {unknown},
series = {Springer INdAM Series},
title = {{Sharpening} of decay rates in {Fourier} based hypocoercivity methods},
url = {https://link.springer.com/chapter/10.1007/978-3-030-82946-9{\_}1},
year = {2021}
}
@unpublished{faucris.287615895,
abstract = {We analyze two recently proposed methods to establish a priori lower bounds
on the minimum of general integral variational problems. The methods, which
involve either 'occupation measures' [Korda et al., ch. 10 of Numerical
Control: Part A, https://doi.org/10.1016/bs.hna.2021.12.010] or a 'pointwise dual relaxation'
procedure [Chernyavsky et al., arXiv:2110.03079], are shown to produce the same
lower bound under a coercivity hypothesis ensuring their strong duality. We
then show by a minimax argument that the methods actually evaluate the minimum
for classes of one-dimensional, scalar-valued, or convex multidimensional
problems. For generic problems, however, we conjecture that these methods
should fail to capture the minimum and produce non-sharp lower bounds. We
explain why using two examples, the first of which is one-dimensional and
scalar-valued with a non-convex constraint, and the second of which is
multidimensional and non-convex in a different way. The latter example
emphasizes the existence in multiple dimensions of nonlinear constraints on
gradient fields that are ignored by occupation measures, but are built into the
finer theory of gradient Young measures, which we review.},
author = {Fantuzzi, Giovanni and Tobasco, Ian},
faupublication = {no},
note = {https://cris.fau.de/converis/publicweb/Publication/287615895},
peerreviewed = {automatic},
title = {{Sharpness} and non-sharpness of occupation measure bounds for integral variational problems},
url = {https://arxiv.org/abs/2207.13570},
year = {2024}
}
@article{faucris.264662466,
abstract = {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.

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).

},
author = {Han, Zhong-Jie and Zuazua Iriondo, Enrique},
faupublication = {yes},
keywords = {Optimal control problems; Riccati operator; slow decay rate; weak controllability and observability; turnpike property},
note = {https://cris.fau.de/converis/publicweb/Publication/263168060},
peerreviewed = {automatic},
title = {{Slow} decay and {Turnpike} for {Infinite}-horizon {Hyperbolic} {LQ} problems},
url = {https://dcn.nat.fau.eu/wp-content/uploads/han-zuazua-infinite-horizon9-2z.pdf},
year = {2024}
}
@article{faucris.243929744,
abstract = {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.},
author = {Monge, Azahar and Zuazua, Enrique},
doi = {10.1016/j.sysconle.2020.104801},
faupublication = {yes},
journal = {Systems & Control Letters},
keywords = {Adjoint problem; Diffusion–advection equation; Inverse problems; Optimal control; Optimization},
note = {CRIS-Team Scopus Importer:2020-10-16},
peerreviewed = {Yes},
title = {{Sparse} source identification of linear diffusion–advection equations by adjoint methods},
volume = {145},
year = {2020}
}
@article{faucris.320058370,
abstract = {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.},
author = {Veldman, Daniel and Borkowski, Alexandra and Zuazua, Enrique},
doi = {10.1109/TAC.2024.3375253},
faupublication = {yes},
journal = {IEEE Transactions on Automatic Control},
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},
note = {CRIS-Team Scopus Importer:2024-03-22},
pages = {1-8},
peerreviewed = {Yes},
title = {{Stability} and {Convergence} of a {Randomized} {Model} {Predictive} {Control} {Strategy}},
year = {2024}
}
@unpublished{faucris.285034893,
abstract = {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.

s(R^{N}) whose energy satisfies [Formula presented]SN,s^{[Formula presented]}≤‖u‖H˙^{s}(R^{N})≤[Formula presented]SN,s^{[Formula presented]}, where SN,s is the optimal Sobolev constant, the bound ‖u−U[z,λ]‖H˙^{s}(R^{N})≲‖(−Δ)^{s}u−u^{2s⁎−1}‖H˙^{−s}(R^{N}), holds for a suitable fractional Talenti bubble U[z,λ]. For functions u which are close to Talenti bubbles, we give the sharp asymptotic value of the implied constant in this inequality. As an application of this, we derive an explicit polynomial extinction rate for positive solutions to a fractional fast diffusion equation.},
author = {De Nitti, Nicola and König, Tobias},
doi = {10.1016/j.jfa.2023.110093},
faupublication = {yes},
journal = {Journal of Functional Analysis},
keywords = {Extinction profiles; Fractional fast diffusion equation; Fractional Sobolev inequality; Stability of critical points},
note = {CRIS-Team Scopus Importer:2023-08-04},
peerreviewed = {Yes},
title = {{Stability} with explicit constants of the critical points of the fractional {Sobolev} inequality and applications to fast diffusion},
volume = {285},
year = {2023}
}
@unpublished{faucris.321081881,
author = {Crin-Barat, Timothée and Kawashima, Shuichi and Xu, Jiang},
faupublication = {yes},
note = {https://cris.fau.de/converis/publicweb/Publication/321081881},
peerreviewed = {automatic},
title = {{The} {Cattaneo}-{Christov} approximation of {Fourier} heat-conductive compressible fluids},
year = {2024}
}
@article{faucris.308878545,
author = {Ftouhi, Ilias and Henrot, Antoine},
faupublication = {yes},
journal = {Mathematical Reports},
peerreviewed = {Yes},
title = {{The} diagram (λ 1 , µ 1 )},
url = {http://imar.ro/journals/Mathematical{\_}Reports/Pdfs/2022/1-2/9.pdf},
year = {2022}
}
@incollection{faucris.253635298,
abstract = {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},
address = {6330 Cham, Switzerland},
author = {Gugat, Martin and Zuazua, Enrique and Schuster, Michael},
booktitle = {Stabilization of Distributed Parameter Systems: Design Methods and Applications},
doi = {10.1007/978-3-030-61742-4},
editor = {Grigory Sklyar, Alexander Zuyev},
faupublication = {yes},
keywords = {Turnpike phenomenon; finite time; optimal control problems;},
pages = {17-41},
peerreviewed = {Yes},
publisher = {Springer Nature Switzerland AG.The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland},
series = {SEMA SIMAI Springer SeriesICIAM 2019 SEMA SIMAI Springer Series},
title = {{The} {Finite}-{Time} {Turnpike} {Phenomenonfor} {Optimal} {Control} {Problems}:{Stabilization} by {Non}-smooth {TrackingTerms}},
volume = {2},
year = {2021}
}
@article{faucris.246701229,
abstract = {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.},
author = {Esteve, Carlos and Zuazua, Enrique},
doi = {10.1137/20M1330130},
faupublication = {yes},
journal = {SIAM Journal on Mathematical Analysis},
keywords = {Hamilton-jacobi equation; Inverse design problem; Obstacle problems; Semiconcave envelopes},
note = {CRIS-Team Scopus Importer:2020-12-11},
pages = {5627-5657},
peerreviewed = {Yes},
title = {{The} inverse problem for {Hamilton}-jacobi equations and semiconcave envelopes},
volume = {52},
year = {2020}
}
@inproceedings{faucris.236484179,
abstract = {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 $u{\_}T$.

We prove that, when the time horizon T tends to infinity, the value function asymptotically behaves as [katex]W(x)+cT+\lambda [/katex], , and we provide a control interpretation of each of these three terms, making clear the link with the turnpike property.

the limit N → ∞ of systems governed by a large number N of ordinary differential equations. We show that the

optimal control problems with large time horizons give rise to a turnpike structure of the optimal state and the

optimal control. For the proof, we use the fact that the turnpike structure for the problems on the level of ordinary

differential equations is preserved under the corresponding mean-field limit.