TY - JOUR A1 - Gugat, Martin A1 - Hante, Falk A1 - Jin, Li T1 - Closed loop control of gas flow in a pipe: Stability for a transient model JF - at - Automatisierungstechnik N2 - This contribution focuses on the analysis and control of friction-dominated flow of gas in pipes. The pressure in the gas flow is governed by a partial differential equation that is a doubly nonlinear parabolic equation of p-Laplace type, where p=2/3. Such equations exhibit positive solutions, finite speed of propagation and satisfy a maximum principle. The pressure is fixed on one end (upstream), and the flow is specified on the other end (downstream). These boundary conditions determine a unique steady equilibrium flow. We present a boundary feedback flow control scheme, that ensures local exponential stability of the equilibrium in an L2-sense. The analysis is done both for the pde system and an ode system that is obtained by a suitable spatial semi-discretization. The proofs are based upon suitably chosen Lyapunov functions. Y1 - 2020 ER - TY - JOUR A1 - Hante, Falk A1 - Leugering, Günter A1 - Martin, Alexander A1 - Schewe, Lars A1 - Schmidt, Martin T1 - Challenges in optimal control problems for gas and fluid flow in networks of pipes and canals: From modeling to industrial applications N2 - 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. KW - Networks KW - pipes KW - optimal control KW - Euler and St. Venant equations KW - hierarchy of models Y1 - 2016 ER - TY - JOUR A1 - Gugat, Martin A1 - Hante, Falk T1 - Lipschitz Continuity of the Value Function in Mixed-Integer Optimal Control Problems JF - Mathematics of Control, Signals, and Systems Y1 - 2017 U6 - https://doi.org/10.1007/s00498-016-0183-4 VL - 29 IS - 1 ER - TY - JOUR A1 - Hante, Falk A1 - Schmidt, Martin T1 - Complementarity-Based Nonlinear Programming Techniques for Optimal Mixing in Gas Networks JF - EURO Journal on Computational Optimization N2 - 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. KW - Gas transport networks KW - Mixing KW - Inverse parametric quadratic programming KW - Complementarity constraints KW - MPCC Y1 - 2017 IS - 7(3) SP - 299 EP - 323 ER - TY - CHAP A1 - Rüffler, Fabian A1 - Hante, Falk T1 - Optimality Conditions for Switching Operator Differential Equations N2 - We consider optimal switching of hybrid abstract evolution equations. The framework includes switching semilinear partial differential equations of parabolic or hyperbolic type, discontinuous state resets, switching costs and allows switching of the principle parts of the equations. We present adjoint-based formulae for the gradient of the cost functional with respect to position and number of switching time points that lead to first order necessary conditions. Moreover, we discuss an alternate-direction approach for implementing descent methods. As an application we consider optimal open/close-switching of valves and on/off-switching control of compressors in a gas network modelled by a graph with simplified euler equations on edges and suitable coupling conditions at nodes. Y1 - 2017 ER - TY - JOUR A1 - Hante, Falk A1 - Mommer, Mario A1 - Potschka, Andreas T1 - Newton-Picard preconditioners for time-periodic, parabolic optimal control problems JF - SIAM Journal on Numerical Analysis Y1 - 2016 U6 - https://doi.org/10.1137/140967969 VL - 53 IS - 5 SP - 2206 EP - 2225 ER - TY - JOUR A1 - Domschke, Pia A1 - Groß, Martin A1 - Hiller, Benjamin A1 - Hante, Falk A1 - Schewe, Lars A1 - Schmidt, Martin T1 - Mathematische Modellierung, Simulation und Optimierung von Gastransportnetzwerken JF - gwf-gas/erdgas Y1 - 2015 VL - 11 SP - 880 EP - 885 ER - TY - CHAP A1 - Hante, Falk T1 - On the relaxation gap for PDE mixed-integer optimal control problems T2 - Proceedings in Applied Mathematics and Mechanics Y1 - 2016 U6 - https://doi.org/10.1002/pamm.201610380 VL - 16 SP - 783 EP - 784 ER - TY - JOUR A1 - Rüffler, Fabian A1 - Hante, Falk T1 - Optimal Switching for Hybrid Semilinear Evolutions JF - Nonlinear Analysis: Hybrid Systems Y1 - 2016 U6 - https://doi.org/10.1016/j.nahs.2016.05.001 VL - 22 SP - 215 EP - 227 ER - TY - JOUR A1 - Gugat, Martin A1 - Leugering, Günter A1 - Hante, Falk ED - Piccoli, Benedetto T1 - Stationary States in Gas Networks JF - Networks and Heterogeneous Media N2 - 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. KW - Network Y1 - 2016 U6 - https://doi.org/doi:10.3934/nhm.2015.10.295 VL - 10 IS - 2 SP - 295 EP - 320 ER - TY - INPR A1 - Hante, Falk T1 - Relaxation methods for optimal switching control of PDE-dynamical systems N2 - The chapter reviews certain computational approaches to solve optimal control problems for evolution-type partial differential equations, where some control functions are limited to switching. The mechanism that enforces switching is modeled as integer restrictions. This brings a combinatorial aspect into the apart from switching already computationally very demanding optimization problems. Recently, great advances have been made to tackle such problems rigorously using relaxation and combinatorial integral approximation. An overview of these methods and the known theoretical results concerning convergence and error estimates are provided in a consistent manner. Further, we point to applications with benchmark character as well as to open problems. Y1 - ER - TY - INPR A1 - Hante, Falk A1 - Krug, Richard A1 - Schmidt, Martin T1 - Time-Domain Decomposition for Mixed-Integer Optimal Control Problems N2 - We consider mixed-integer optimal control problems, whose optimality conditions involve global combinatorial optimization aspects for the corresponding Hamiltonian pointwise in time. We propose a time-domain decomposition, which makes this problem class accessible for mixed-integer programming using parallel-in-time direct discretizations. The approach is based on a decomposition of the optimality system and the interpretation of the resulting subproblems as suitably chosen mixed-integer optimal control problems on subintervals in time. An iterative procedure then ensures continuity of the states at the boundaries of the subintervals via co-state information encoded in virtual controls. We prove convergence of this iterative scheme for discrete-continuous linear-quadratic problems and present numerical results both for linear-quadratic as well as nonlinear problems. KW - Mixed-integer optimal control problems KW - Time-domain decomposition KW - Mixed-integer nonlinear optimization KW - Convergence Y1 - 2021 ER - TY - JOUR A1 - Hante, Falk T1 - Mixed-Integer Optimal Control for PDEs: Relaxation via Differential Inclusions and Applications to Gas Network Optimization JF - Mathematical Modelling, Optimization, Analytic and Numerical Solutions, Springer Series on Industrial and Applied Mathematics, 2020 N2 - We show that mixed-integer control problems for evolution type partial differential equations can be regarded as operator differential inclusions. This yields a relaxation result including a characterization of the optimal value for mixed-integer optimal control problems with control constraints. The theory is related to partial outer convexification and sum-up rounding methods. The results are applied to optimal valve switching control for gas pipeline operations. A numerical example illustrates the approach. Y1 - 2018 ER - TY - JOUR A1 - Hante, Falk M. A1 - Schmidt, Martin T1 - Convergence of Finite-Dimensional Approximations for Mixed-Integer Optimization with Differential Equations JF - Control and Cybernetics N2 - We consider a direct approach to solve mixed-integer 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 maximum step size for discretizing 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. KW - Optimization with differential equations KW - Optimal value function KW - Lipschitz continuity KW - Parametric optimization KW - Mixed-integer nonlinear programming Y1 - 2018 ER - TY - JOUR A1 - Gugat, Martin A1 - Hante, Falk T1 - On the turnpike phenomenon for optimal boundary control problems with hyperbolic systems JF - SIAM Journal on Control and Optimization N2 - 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 to 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. Y1 - 2019 U6 - https://doi.org/10.1137/17M1134470 VL - 57 IS - 1 SP - 264 EP - 289 ER - TY - INPR A1 - Brodskyi, Yan A1 - Hante, Falk A1 - Seidel, Arno T1 - Stabilization of Partial Differential Equations by Sequential Action Control N2 - We extend the framework of sequential action control to systems of partial differential equations which can be posed as abstract linear control problems in a Hilbert space. We follow a late-lumping approach and show that the control action can be explicitly obtained from variational principles using adjoint information. Moreover, we analyze the closed-loop system obtained from the SAC feedback for quadratic stage costs. We apply this theory prototypically to an unstable heat equation and verify the results numerically. Y1 - 2019 ER - TY - JOUR A1 - Göttlich, Simone A1 - Hante, Falk A1 - Potschka, Andreas A1 - Schewe, Lars T1 - Penalty alternating direction methods for mixed-integer optimal control with combinatorial constraints JF - Mathematical Programming N2 - We consider mixed-integer optimal control problems with combinatorial constraints that couple over time such as minimum dwell times. We analyze a lifting and decomposition approach into a mixed-integer optimal control problem without combinatorial constraints and a mixed-integer problem for the combinatorial constraints in the control space. Both problems can be solved very efficiently with existing methods such as outer convexification with sum-up-rounding strategies and mixed-integer linear programming techniques. The coupling is handled using a penalty-approach. We provide an exactness result for the penalty which yields a solution approach that convergences to partial minima. We compare the quality of these dedicated points with those of other heuristics amongst an academic example and also for the optimization of electric transmission lines with switching of the network topology for flow reallocation in order to satisfy demands. Y1 - 2019 ER - TY - JOUR A1 - Rüffler, Fabian A1 - Mehrmann, Volker A1 - Hante, Falk T1 - Optimal Model Switching for Gas Flow in Pipe Networks N2 - We consider model adaptivity for gas flow in pipeline networks. For each instant in time and for each pipe in the network a model for the gas flow is to be selected from a hierarchy of models in order to maximize a performance index that balances model accuracy and computational cost for a simulation of the entire network. This combinatorial problem involving partial differential equations is posed as an optimal switching control problem for abstract semilinear evolutions. We provide a theoretical and numerical framework for solving this problem using a two stage gradient descent approach based on switching time and mode insertion gradients. A numerical study demonstrates the practicability of the approach. Y1 - 2018 ER - TY - INPR A1 - Hante, Falk M. A1 - Schmidt, Martin T1 - Gas Transport Network Optimization: Mixed-Integer Nonlinear Models N2 - Although modern societies strive towards energy systems that are entirely based on renewable energy carriers, natural gas is still one of the most important energy sources. This became even more obvious in Europe with Russia's 2022 war against the Ukraine and the resulting stop of gas supplies from Russia. Besides that it is very important to use this scarce resource efficiently. To this end, it is also of significant relevance that its transport is organized in the most efficient, i.e., cost- or energy-efficient, way. The corresponding mathematical optimization models have gained a lot of attention in the last decades in different optimization communities. These models are highly nonlinear mixed-integer problems that are constrained by algebraic constraints and partial differential equations (PDEs), which usually leads to models that are not tractable. Hence, simplifications have to be made and in this chapter, we present a commonly accepted finite-dimensional stationary model, i.e., a model in which the steady-state solutions of the PDEs are approximated with algebraic constraints. For more details about the involved PDEs and the treatment of transient descriptions we refer to Hante and Schmidt (2023). The presented finite-dimensional as well as mixed-integer nonlinear and nonconvex model is still highly challenging if it needs to be solved for real-world gas transport networks. Hence, we also review some classic solution approaches from the literature. KW - Gas networks KW - Mixed-integer nonlinear optimization KW - Mixed-integer linear optimization KW - Nonlinear optimization Y1 - 2023 ER - TY - INPR A1 - Hante, Falk A1 - Kuchler, Christian T1 - Indirect methods for optimal control of hybrid PDE-dynamical / switching systems using relaxation N2 - We propose a novel algorithmic approach to computationally solve optimal control problems governed by linear evolution-type PDEs including a state-dependent control-regime switching mechanism. We introduce an equivalent mixed-integer formulation featuring vanishing constraints arising by methods of disjunctive programming. We embed the problem into the class of equilibrium constraints by introduction of an additional slack variable. Based on theoretical results associated with Sum-Up-Rounding strategies, we proceed with the solution of the related relaxed formulation by an indirect approach. In order to obtain a computationally tractable optimality system, we apply a Moreau-Yosida type penalty approach of the vanishing constraints. After the theoretical discussion, we introduce and exert the algorithmic framework founded on a semismooth Newton method. Finally, we communicate computational experiments based on our approach. Y1 - ER - TY - INPR A1 - Hante, Falk M. A1 - Schmidt, Martin T1 - Gas Transport Network Optimization: PDE-Constrained Models N2 - The optimal control of gas transport networks was and still is a very important topic for modern economies and societies. Accordingly, a lot of research has been carried out on this topic during the last years and decades. Besides mixed-integer aspects in gas transport network optimization, one of the main challenges is that a physically and technically detailed modeling of transient gas dynamics leads to theoretically and computationally highly demanding models involving nonlinear partial differential equations (PDEs). For further background on the application, historical notes and a detailed discussion of mixed-integer aspects for stationary descriptions we refer to Hante and Schmidt (2023). In this chapter, we focus on the most common modeling approaches concerning transient descriptions, point out the challenges, and summarize important contributions concerning the optimization of the most relevant control parameters for this particular class of problems. KW - Gas networks KW - Partial differential equations KW - Optimal control KW - PDE-constrained optimization KW - Modeling Y1 - 2023 ER - TY - INPR A1 - Hante, Falk A1 - Kuchler, Christian T1 - An Algorithmic Framework for Optimal Control of Hybrid Dynamical System with Parabolic PDEs N2 - We present an algorithmic approach for the computational solution of optimal control problems with hybrid nature governed by linear parabolic PDEs featuring implicit switches. We propose a stepwise reformulation of the original formulation into a more tractable setting via application of methods from disjunctive programming and a time transformation method. After removal of the implicit switching rule at the cost of the introduction of explicit switching variables and vanishing constraints, the connection of the resulting formulation to problems with equilibrium constraints is established and studied. The previous steps in combination with smoothening and a Moreau-Yosida type penalty approach allow the derivation of necessary first order optimality conditions to characterize candidates for optimality to the original system. Following the discussion of each individual reformulation step, we introduce the algorithmic framework founded on a semismooth Newton method. Finally, we report on computational of the proposed framework. Y1 - 2023 ER - TY - INPR A1 - Hante, Falk M. A1 - Schmidt, Martin A1 - Topalovic, Antonia T1 - Stabilizing GNEP-Based Model Predictive Control: Quasi-GNEPs and End Constraints N2 - We present a feedback scheme for non-cooperative dynamic games and investigate its stabilizing properties. The dynamic games are modeled as generalized Nash equilibrium problems (GNEP), in which the shared constraint consists of linear time-discrete dynamic equations (e.g., sampled from a partial or ordinary differential equation), which are jointly controlled by the players’ actions. Further, the individual objectives of the players are interdependent and defined over a fixed time horizon. The feedback law is synthesized by moving-horizon model predictive control (MPC). We investigate the asymptotic stability of the resulting closed-loop dynamics. To this end, we introduce α-quasi GNEPs, a family of auxiliary problems based on a modification of the Nikaido–Isoda function, which approximate the original games. Basing the MPC scheme on these auxiliary problems, we derive conditions on the players’ objectives, which guarantee asymptotic stability of the closed-loop if stabilizing end constraints are enforced. This analysis is based on showing that the associated optimal-value function is a Lyapunov function. Additionally, we identify a suitable Lyapunov function for the MPC scheme based on the original GNEP, whose solution fulfills the stabilizing end constraints. The theoretical results are complemented by numerical experiments. KW - Model predictive control KW - Non-cooperative distributed control KW - Closed-loop stability KW - Generalized Nash equilibrium problems Y1 - 2024 ER -