An augmented BV setting for feedback switching control - Hante, Falk; Leugering, Günter; Seidman, Thomas I. (Journal of Systems Science & Complexity, 2010)
This paper considers dynamical systems under feedback with control actions limited to switching. The authors wish to understand the closed-loop systems as approximating multi-scale problems in which the implementation of switching merely acts on a fast scale. Such hybrid dynamical systems are extensively studied in the literature, but not much so far for feedback with partial state observation. This becomes in particular relevant when the dynamical systems are governed by partial differential equations. The authors introduce an augmented BV setting which permits recognition of certain fast scale effects and give a corresponding well-posedness result for observations with such minimal regularity. As an application for this setting, the authors show existence of solutions for systems of semilinear hyperbolic equations under such feedback with pointwise observations.

Challenges in Optimal Control Problems for Gas and Fluid Flow in Networks of Pipes and Canals: From Modeling to Industrial Applications - Hante, Falk; Leugering, Günter; Martin, Alexander; Schewe, Lars; Schmidt, Martin (Industrial Mathematics and Complex Systems: Emerging Mathematical Models, Methods and Algorithms, 2017)
We consider optimal control problems for the flow of gas or fresh water in pipe networks as well as drainage or sewer systems in open canals. The equations of motion are taken to be represented by the nonlinear isothermal Euler gas equations, the water hammer equations, or the St. Venant equations for flow. We formulate model hierarchies and derive an abstract model for such network flow problems including pipes, junctions, and controllable elements such as valves, weirs, pumps, as well as compressors. We use the abstract model to give an overview of the known results and challenges concerning equilibria, well-posedness, controllability, and optimal control. A major challenge concerning the optimization is to deal with switching on-off states that are inherent to controllable devices in such applications combined with continuous simulation and optimization of the gas flow. We formulate the corresponding mixed-integer nonlinear optimal control problems and outline a decomposition approach as a solution technique.
Keywords: Networks; pipes; canals; Euler and St. Venant equations; hierarchy of models; domain decomposition; controllability; optimal control.

Complementarity-Based Nonlinear Programming Techniques for Optimal Mixing in Gas Networks - Hante, Falk; Schmidt, Martin (2017)
We consider nonlinear and nonsmooth mixing aspects in gas transport optimization problems. As mixed-integer reformulations of pooling-type mixing models already render small-size instances computationally intractable, we investigate the applicability of smooth nonlinear programming techniques for equivalent complementarity-based reformulations. Based on recent results for remodeling piecewise affine constraints using an inverse parametric quadratic programming approach, we show that classical stationarity concepts are meaningful for the resulting complementarity-based reformulation of the mixing equations. Further, we investigate in a numerical study the performance of this reformulation compared to a more compact complementarity-based one that does not feature such beneficial regularity properties. All computations are performed on publicly available data of real-world size problem instances from steady-state gas transport.
Keywords: Gas transport networks; Mixing; Inverse parametric quadratic programming; Complementarity constraints; MPCC

Complementarity-based nonlinear programming techniques for optimal mixing in gas networks - Hante, Falk; Schmidt, Martin (EURO Journal on Computational Optimization, 2019)
We consider nonlinear and nonsmooth mixing aspects in gas transport optimization problems. As mixed-integer reformulations of pooling-type mixing models already render small-size instances computationally intractable, we investigate the applicability of smooth nonlinear programming techniques for equivalent complementarity-based reformulations. Based on recent results for remodeling piecewise affine constraints using an inverse parametric quadratic programming approach, we show that classical stationarity concepts are meaningful for the resulting complementarity-based reformulation of the mixing equations. Further, we investigate in a numerical study the performance of this reformulation compared to a more compact complementarity-based one that does not feature such beneficial regularity properties. All computations are performed on publicly available data of real-world size problem instances from steady-state gas transport.
CRIS-Team Scopus Importer:2019-06-18
Keywords: Complementarity constraints; Gas transport networks; Inverse parametric quadratic programming; Mixing; MPCC

Convergence of finite-dimensional approximations for mixed-integer optimization with differential equations∗ - Hante, Falk; Schmidt, Martin (Control and Cybernetics, 2019)
We consider a direct approach to solving the mixedinteger nonlinear optimization problems with constraints depending on initial and terminal conditions of an ordinary differential equation. In order to obtain a finite-dimensional problem, the dynamics are approximated using discretization methods. In the framework of general one-step methods, we provide sufficient conditions for the convergence of this approach in the sense of the corresponding optimal values. The results are obtained by considering the discretized problem as a parametric mixed-integer nonlinear optimization problem in finite dimensions, where the step size for discretization of the dynamics is the parameter. In this setting, we prove the continuity of the optimal value function under a stability assumption for the integer feasible set and second-order conditions from nonlinear optimization. We address the necessity of the conditions on the example of pipe sizing problems for gas networks.
CRIS-Team Scopus Importer:2021-04-20
Keywords: Lipschitz continuity; Mixedinteger nonlinear programming; Optimal value function; Optimization with differential equations; Parametric optimization

Converse Lyapunov theorems for switched systems in Banach and Hilbert spaces - Hante, Falk; Sigalotti, Mario (SIAM Journal on Control and Optimization, 2011)

Existence of common Lyapunov functions for infinite-dimensional switched linear systems - Hante, Falk; Sigalotti, Mario (Decision and Control (CDC) 2010. CDC 2010, 49th IEEE Conference on, 2010)

Exponential stability of switched linear hyperbolic initial-boundary value problems - Hante, Falk; Amin, Saurabh; Bayen, Alexandre M. (IEEETransactions on Automatic Control, 2012)

Hybrid Dynamics Comprising Modes Governed by Partial Differential Equations: Modeling, Analysis and Control for Semilinear Hyperbolic Systems in One Space Dimension - Hante, Falk (Friedrich-Alexander-Universität Erlangen-Nürnberg, 2010)

Lipschitz Continuity of the Value Function in Mixed-Integer Optimal Control Problems - Gugat, Martin; Hante, Falk (Mathematics of Control Signals and Systems, 2017)
Keywords: Parametric optimal control; Parametric switching control; Parametric optimization; Sensitivity; Mixed-integer optimal control problems; Optimal value function; Lipschitz continuity

Mathematische Modellierung, Simulation und Optimierung von Gastransportnetzwerken - Domschke, Pia; Groß, Martin; Hante, Falk; Hiller, Benjamin; Schewe, Lars; Schmidt, Martin (Gas und Wasserfach, Gas, Erdgas, 2015)

Model-Based Design of Biochemical Microreactors - Elbinger, Tobias; Gahn, Markus; Hante, Falk; Voll, Lars; Leugering, Günter; Knabner, Peter; Neuss-Radu, Maria (Frontiers in Bioengineering and Biotechnology, 2016)
Mathematical modeling of biochemical pathways is an important resource in Synthetic Biology, as the predictive power of simulating synthetic pathways represents an important step in the design of synthetic metabolons. In this paper, we are concerned with the mathematical modeling, simulation, and optimization of metabolic processes in biochemical microreactors able to carry out enzymatic reactions and to exchange metabolites with their surrounding medium. The results of the reported modeling approach are incorporated in the design of the first microreactor prototypes that are under construction. These microreactors consist of compartments separated by membranes carrying specific transporters for the input of substrates and export of products. Inside the compartments of the reactor multienzyme complexes assembled on nano-beads by peptide adapters are used to carry out metabolic reactions. The spatially resolved mathematical model describing the ongoing processes consists of a system of diffusion equations together with boundary and initial conditions. The boundary conditions model the exchange of metabolites with the neighboring compartments and the reactions at the surface of the nano-beads carrying the multienzyme complexes. Efficient and accurate approaches for numerical simulation of the mathematical model and for optimal design of the microreactor are developed. As a proof-of-concept scenario, a synthetic pathway for the conversion of sucrose to glucose-6-phosphate (G6P) was chosen. In this context, the mathematical model is employed to compute the spatio-temporal distributions of the metabolite concentrations, as well as application relevant quantities like the outflow rate of G6P. These computations are performed for different scenarios, where the number of beads as well as their loading capacity are varied. The computed metabolite distributions show spatial patterns, which differ for different experimental arrangements. Furthermore, the total output of G6P increases for scenarios where microcompartimentation of enzymes occurs. These results show that spatially resolved models are needed in the description of the conversion processes. Finally, the enzyme stoichiometry on the nano-beads is determined, which maximizes the production of glucose-6-phosphat

Modeling and Analysis of Modal Switching in Networked Transport Systems - Hante, Falk; Leugering, Günter; Seidman, Thomas I. (Applied Mathematics and Optimization, 2009)
We consider networked transport systems defined on directed graphs: the dynamics on the edges correspond to solutions of transport equations with space dimension one. In addition to the graph setting, a major consideration is the introduction and propagation of discontinuities in the solutions when the system may discontinuously switch modes, naturally or as a hybrid control. This kind of switching has been extensively studied for ordinary differential equations, but not much so far for systems governed by partial differential equations. In particular, we give well-posedness results for switching as a control, both in finite horizon open loop operation and as feedback based on sensor measurements in the system. © 2008 Springer Science+Business Media, LLC.
UnivIS-Import:2015-03-09:Pub.2009.nat.dma.lama1.modeli
Keywords: Automated switching; Hybrid system; Networked system; Optimal control; Transport equation

Newton-Picard preconditioners for time-periodic parabolic optimal control problems - Hante, Falk; Mommer, Mario; Potschka, Andreas (SIAM Journal on Numerical Analysis, 2015)

On conditions for asymptotic stability of dissipative infinite-dimensional systems with intermittent damping - Hante, Falk; Sigalotti, Mario; Tucsnak, Marius (Journal of Differential Equations, 2012)

On stability of switched linear hyperbolic conservation laws with reflecting boundaries - Hante, Falk; Amin, Saurabh; Bayen, Alexandre M. (Hybrid systems: computation and control, 2008)

On the relaxation gap for PDE mixed-integer optimal control problems - Hante, Falk (Proceedings in Applied Mathematics and Mechanics, 2016)

ON THE TURNPIKE PHENOMENON FOR OPTIMAL BOUNDARY CONTROL PROBLEMS WITH HYPERBOLIC SYSTEMS - Gugat, Martin; Hante, Falk (SIAM Journal on Control and Optimization, 2019)
We study problems of optimal boundary control with systems governed by linear hyperbolic partial differential equations. The objective function is quadratic and given by an integral over the finite time interval (0, T) that depends on the boundary traces of the solution. If the time horizon T is sufficiently large, the solution of the dynamic optimal boundary control problem can be approximated by the solution of a steady state optimization problem. We show that for T -> infinity the approximation error converges to zero in the sense of the norm in L-2 (0, 1) with the rate 1/T, if the time interval (0, T) is transformed to the fixed interval (0, 1). Moreover, we show that also for optimal boundary control problems with integer constraints for the controls the turnpike phenomenon occurs. In this case the steady state optimization problem also has the integer constraints. If T is sufficiently large, the integer part of each solution of the dynamic optimal boundary control problem with integer constraints is equal to the integer part of a solution of the static problem. A numerical verification is given for a control problem in gas pipeline operations.
CRIS-Team WoS Importer:2019-03-15

Optimal Boundary Control of Convection-Reaction Transport Systems with Binary Control Functions - Hante, Falk; Leugering, Günter (Lecture Notes in Computer Science, 2009)
We investigate a new approach for solving boundary control problems for dynamical systems that are governed by transport equations, when the control function is restricted to binary values. We consider these problems as hybrid dynamical systems embedded with partial differential equations and present an optimality condition based on sensitivity analysis for the objective when the dynamics are governed by semilinear convection-reaction equations. These results make the hybrid problem accessible for continuous non-linear optimization techniques. For the computation of optimal solution approximations, we propose using meshfree solvers to overcome essential difficulties with numerical dissipation for these distributed hybrid systems. We compare results obtained by the proposed method with solutions taken from a mixed inter programming formulation of the control problem.
UnivIS-Import:2015-04-16:Pub.2009.nat.dma.lama1.optima

Optimal switching for hybrid semilinear evolutions - Rüffler, Fabian; Hante, Falk (Nonlinear Analysis: Hybrid Systems, 2016)
Keywords: Partial differential equation

Relaxation methods for mixed-integer optimal control of partial differential equations - Hante, Falk; Sager, Sebastian (Computational Optimization and Applications, 2013)

Stability analysis of linear hyperbolic systems with switching parameters and boundary conditions - Hante, Falk; Amin, Saurabh; Bayen, Alexandre M. (Decision and Control, 2008. CDC 2008, 47th IEEE Conference on, 2008)

STATIONARY STATES IN GAS NETWORKS - Gugat, Martin; Hante, Falk; Hirsch-Dick, Markus; Leugering, Günter (Networks and Heterogeneous Media, 2015)
Pipeline networks for gas transportation often contain circles. For such networks it is more difficult to determine the stationary states than for networks without circles. We present a method that allows to compute the stationary states for subsonic pipe flow governed by the isothermal Euler equations for certain pipeline networks that contain circles. We also show that suitably chosen boundary data determine the stationary states uniquely. The construction is based upon novel explicit representations of the stationary states on single pipes for the cases with zero slope and with nonzero slope. In the case with zero slope, the state can be represented using the Lambert-W function.
Keywords: Network; isothermal Euler equations; stationary state; circles; cycles; gas transportation network