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 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 - 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 - JOUR A1 - Schuster, Michael A1 - Gugat, Martin A1 - Sokolowski, Jan T1 - The Location Problem for Compressor Stations in Pipeline Networks JF - Mathematics and Mechanics of Complex Systems N2 - In the operation of pipeline networks, compressors play a crucial role in ensuring the network’s functionality for various scenarios. In this contribution we address the important question of finding the optimal location of the compressors. This problem is of a novel structure, since it is related to the gas dynamics that governs the network flow. That results in nonconvex mixed integer stochastic optimization problems with probabilistic constraints. Using a steady state model for the gas flow in pipeline networks including compressor control and uncertain loads given by certain probability distributions, we consider the problem of finding the optimal location for the control on the network such that the control cost is minimal and the gas pressure stays within given bounds. In the deterministic setting, we present explicit bounds for the pipe length and the inlet pressure such that a unique optimal compressor location with minimal control cost exists. In the probabilistic setting, we give an existence result for the optimal compressor location and discuss the uniqueness of the solution depending on the probability distribution. For Gaussian distributed loads a uniqueness result for the optimal compressor location is presented. We further present the problem of finding optimal compressor locations on networks including the number of compressor stations as a variable. Results for the existence of optimal locations on a graph in both the deterministic and the probabilistic setting are presented, and the uniqueness of the solutions is discussed depending on probability distributions and graph topology. The paper concludes with an illustrative example on a diamond graph demonstrating that the minimal number of compressor stations is not necessarily equal to the optimal number of compressor stations. KW - gas network KW - compressor control KW - Weber problem KW - uncertain boundary data KW - non convex mixed integer stochastic problem Y1 - 2024 U6 - https://doi.org/10.2140 VL - 12 IS - 4 SP - 507 EP - 546 ER - TY - JOUR A1 - Gugat, Martin A1 - Leugering, Günter A1 - Wang, Ke ED - Zhang, Xu T1 - Neumann boundary feedback stabilization for a nonlinear wave equation: A strict H2-Lyapunov function JF - Mathematical Control and Related Fields (MCRF) N2 - For a system that is governed by the isothermal Euler equations with friction for ideal gas, the corresponding field of characteristic curves is determined by the velocity of the flow. This velocity is determined by a second-order quasilinear hyperbolic equation. For the corresponding initial-boundary value problem with Neumann-boundary feedback, we consider non-stationary solutions locally around a stationary state on a finite time interval and discuss the well-posedness of this kind of problem. We introduce a strict H2-Lyapunov function and show that the boundary feedback constant can be chosen such that the H2-Lyapunov function and hence also the H2-norm of the difference between the non-stationary and the stationary state decays exponentially with time. KW - Boundary feedback control, feedback stabilization, exponential stability, isothermal Euler equations, second-order quasilinear equation, Lyapunov function, stationary state, non-stationary state, gas pipeline. Y1 - 2017 U6 - https://doi.org/10.3934/mcrf.2017015 VL - 7 IS - 3 SP - 419 EP - 448 ER - TY - JOUR A1 - Gugat, Martin A1 - Leugering, Günter ED - Zuazua, Enrique T1 - Time delay in optimal control loops for wave equations JF - ESAIM: COCV N2 - In optimal control loops delays can occur, for example through transmission via digital communication channels. Such delays influence the state that is generated by the implemented control. We study the effect of a delay in the implementation of L 2-norm minimal Neumann boundary controls for the wave equation. The optimal controls are computed as solutions of problems of exact optimal control, that is if they are implemented without delay, they steer the system to a position of rest in a given finite time T. We show that arbitrarily small delays δ > 0 can have a destabilizing effect in the sense that we can find initial states such that if the optimal control u is implemented in the form yx(t, 1) = u(t − δ) for t > δ, the energy of the system state at the terminal time T is almost twice as big as the initial energy. We also show that for more regular initial states, the effect of a delay in the implementation of the optimal control is bounded above in the sense that for initial positions with derivatives of BV-regularity and initial velocities with BV-regularity, the terminal energy is bounded above by the delay δ multiplied with a factor that depends on the BV-norm of the initial data. We show that for more general hyperbolic optimal exact control problems the situation is similar. For systems that have arbitrarily large eigenvalues, we can find terminal times T and arbitrarily small time delays δ, such that at the time T + δ, in the optimal control loop with delay the norm of the state is twice as large as the corresponding norm for the initial state. Moreover, if the initial state satisfies an additional regularity condition, there is an upper bound for the effect of time delay of the order of the delay with a constant that depends on the initial state only. KW - PDE constrained optimization KW - delay KW - wave equation KW - boundary control KW - hyperbolic system Y1 - 2016 U6 - https://doi.org/http://dx.doi.org/10.1051/cocv/2015038 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 -