@article{GugatLeugeringMartinetal.2018, author = {Gugat, Martin and Leugering, G{\"u}nter and Martin, Alexander and Schmidt, Martin and Sirvent, Mathias and Wintergerst, David}, title = {Towards Simulation Based Mixed-Integer Optimization with Differential Equations}, series = {Networks}, journal = {Networks}, doi = {10.1002/net.21812}, pages = {24}, year = {2018}, abstract = {We propose a decomposition based method for solving mixed-integer nonlinear optimization problems with "black-box" nonlinearities, where the latter, e.g., may arise due to differential equations or expensive simulation runs. The method alternatingly solves a mixed-integer linear master problem and a separation problem for iteratively refining the mixed-integer linear relaxation of the nonlinear equalities. The latter yield nonconvex feasible sets for the optimization model but we have to restrict ourselves to convex and monotone constraint functions. Under these assumptions, we prove that our algorithm finitely terminates with an approximate feasible global optimal solution of the mixed integer nonlinear problem. Additionally, we show the applicability of our approach for three applications from optimal control with integer variables, from the field of pressurized flows in pipes with elastic walls, and from steady-state gas transport. For the latter we also present promising numerical results of our method applied to real-world instances that particularly show the effectiveness of our method for problems defined on networks.}, language = {en} } @article{GugatLeugeringMartinetal.2017, author = {Gugat, Martin and Leugering, G{\"u}nter and Martin, Alexander and Schmidt, Martin and Sirvent, Mathias and Wintergerst, David}, title = {MIP-Based Instantaneous Control of Mixed-Integer PDE-Constrained Gas Transport Problems}, series = {Computational Optimization and Applications}, volume = {70}, journal = {Computational Optimization and Applications}, number = {1}, doi = {10.1007/s10589-017-9970-1}, pages = {267 -- 294}, year = {2017}, abstract = {We study the transient optimization of gas transport networks including both discrete controls due to switching of controllable elements and nonlinear fluid dynamics described by the system of isothermal Euler equations, which are partial differential equations in time and 1-dimensional space. This combination leads to mixed-integer optimization problems subject to nonlinear hyperbolic partial differential equations on a graph. We propose an instantaneous control approach in which suitable Euler discretizations yield systems of ordinary differential equations on a graph. This networked system of ordinary differential equations is shown to be well-posed and affine-linear solutions of these systems are derived analytically. As a consequence, finite-dimensional mixed-integer linear optimization problems are obtained for every time step that can be solved to global optimality using general-purpose solvers. We illustrate our approach in practice by presenting numerical results on a realistic gas transport network.}, language = {en} } @unpublished{GugatKrugMartin2021, author = {Gugat, Martin and Krug, Richard and Martin, Alexander}, title = {Transient gas pipeline flow: Analytical examples, numerical simulation and a comparison to the quasi-static approach}, doi = {10.1007/s11081-021-09690-4}, year = {2021}, abstract = {The operation of gas pipeline flow with high pressure and small Mach numbers allows to model the flow by a semilinear hyperbolic system of partial differential equations. In this paper we present a number of transient and stationary analytical solutions of this model. They are used to discuss and clarify why a pde model is necessary to handle certain dynamic situations in the operation of gas transportation networks. We show that adequate numerical discretizations can capture the dynamical behavior sufficiently accurate. We also present examples that show that in certain cases an optimization approach that is based upon multi-period optimization of steady states does not lead to approximations that converge to the optimal state.}, language = {en} } @article{GugatLazar2023, author = {Gugat, Martin and Lazar, Martin}, title = {Turnpike Properties for Partially Uncontrollable Systems}, volume = {Automatica}, number = {149}, year = {2023}, abstract = {We analyse the turnpike properties for a general, infinite dimensional, linear-quadratic (LQ) optimal control problem, both in the deterministic and in the stochastic case. The novelty of the paper is twofold. Firstly, it obtains positive turnpike results for systems that are (partially) uncontrollable. Secondly, it provides turnpike results for averaged control associated to a family of problems that depend on a random parameter, which is the first turnpike type result in the averaged controllability framework.}, language = {en} } @unpublished{GugatHabermannHintermuelleretal.2021, author = {Gugat, Martin and Habermann, Jens and Hinterm{\"u}ller, Michael and Huber, Olivier}, title = {Constrained exact boundary controllability of a semilinear model for pipeline gas flow}, year = {2021}, abstract = {While the quasilinear isothermal Euler equations are an excellent model for gas pipeline flow, the operation of the pipeline flow with high pressure and small Mach numbers allows us to obtain approximate solutions by a simpler semilinear model. We provide a derivation of the semilinear model that shows that the semilinear model is valid for sufficiently low Mach numbers and sufficiently high pressures. We prove an existence result for continuous solutions of the semilinear model that takes into account lower and upper bounds for the pressure and an upper bound for the magnitude of the Mach number of the gas flow. These state constraints are important both in the operation of gas pipelines and to guarantee that the solution remains in the set where the model is physically valid. We show the constrained exact boundary controllability of the system with the same pressure and Mach number constraints.}, language = {en} } @article{MartinGiesselmannKunkel2021, author = {Martin, Gugat and Giesselmann, Jan and Kunkel, Teresa}, title = {Exponential synchronization of a nodal observer for a semilinear model for the flow in gas networks}, address = {IMA Journal of Mathematical Control and Information}, doi = {10.1093/imamci/dnab029}, year = {2021}, abstract = {The flow of gas through networks of pipes can be modeled by coupling hyperbolic systems of partial differential equations that describe the flow through the pipes that form the edges of the graph of the network by algebraic node conditions that model the flow through the vertices of the graph. In the network, measurements of the state are available at certain points in space.Based upon these nodal observations, the complete system state can be approximated using an observer system. In this paper we present a nodal observer, and prove that the state of the observer system converges to the original state exponentially fast. Numerical experiments confirm the theoretical findings.}, language = {en} } @article{GugatHanteJin2020, author = {Gugat, Martin and Hante, Falk and Jin, Li}, title = {Closed loop control of gas flow in a pipe: Stability for a transient model}, series = {at - Automatisierungstechnik}, journal = {at - Automatisierungstechnik}, year = {2020}, abstract = {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.}, language = {en} } @unpublished{GugatHerty2020, author = {Gugat, Martin and Herty, Michael}, title = {Modeling, Control and Numerics of Gas Networks}, year = {2020}, abstract = {In this article 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.}, language = {en} } @article{SchusterStrauchGugatetal.2020, author = {Schuster, Michael and Strauch, Elisa and Gugat, Martin and Lang, Jens}, title = {Probabilistic Constrained Optimization on Flow Networks}, volume = {Optimization and Engineering}, doi = {https://doi.org/10.1007/s11081-021-09619-x}, pages = {50}, year = {2020}, 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 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.}, language = {en} } @article{GugatSchusterZuazua2020, author = {Gugat, Martin and Schuster, Michael and Zuazua, Enrique}, title = {M. Gugat, M. Schuster, E. Zuazua. The Finite-Time Turnpike Phenomenon for Optimal Control Problems: Stabilization by Non-Smooth Tracking Terms, in "Stabilization of Distributed Parameter Systems: Design Methods and Applications". Grigory Sklyar Alexander Zuyev Eds., ICIAM 2019 SEMA SIMAI Springer Series 2, p. 17-42. ISSN 2199-3041}, volume = {SEMA SIMAI Springer Series 2}, edition = {Grigory Sklyar Alexander Zuyev Eds., ICIAM 2019}, publisher = {Springer International Publishing}, pages = {17 -- 42}, year = {2020}, 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 time.}, language = {en} } @article{GugatGiesselmann2020, author = {Gugat, Martin and Giesselmann, Jan}, title = {Boundary feedback stabilization of a semilinear model for the flow in star-shaped gas networks}, address = {ESAIM:COCV}, doi = {10.1051/cocv/2021061}, year = {2020}, abstract = {The flow of gas through a pipeline network can be modelled by a coupled system of 1-d quasilinear hyperbolic equations. In this system, the influence of certain source terms that model friction effects is essential. Often for the solution of control problems it is convenient to replace the quasilinear model by a simpler semilinear model. In this paper, we analyze the behavior of such a semilinear model on a star-shaped network. The model is derived from the diagonal form of the quasilinear model by replacing the eigenvalues by the sound speed multiplied by 1 or -1 respectively. Thus in the corresponding eigenvalues the influence of the gas velocity is neglected, which is justified in the applications since it is much smaller than the sound speed in the gas. For a star-shaped network of horizontal pipes for suitable coupling conditions we present boundary feedback laws that stabilize the system state exponentially fast to a position of rest for sufficiently small initial data. We show the exponential decay of the \$H^1\$-norm for arbitrarily long pipes. This is remarkable since in general even for linear systems, for certain source terms the system can become exponentially unstable if the space interval is too long. Our proofs are based upon observability inequalities for the \$L^2\$ and the \$H^1\$-norm.}, language = {en} } @unpublished{GugatHerty2020, author = {Gugat, Martin and Herty, Michael}, title = {Limits of stabilizabilizy for a semilinear model for gas pipeline flow}, year = {2020}, abstract = {We present a positive and a negative stabilization result for a semilinear model of gas flow in pipelines. For feedback boundary conditions we obtain an unconditional stabilization result 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}, language = {en} } @unpublished{GugatSokolowski, author = {Gugat, Martin and Sokolowski, Jan}, title = {On Problems of Dynamic Optimal Nodal control for Gas Networks}, abstract = {We consider a dynamic ptimal control problem for gas pipeline systems. The flow is governed by a quasilinear hyperbolic model. Since in the operation of the gas networks regular solutions without shocks are desirable, we impose appropriate state and control constraint in order to guarantee that a classical solution is generated. Due to a W^{2;inf}-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 suffciently large.}, language = {en} } @article{Gugat2021, author = {Gugat, Martin}, title = {On the turnpike property with interior decay for optimal control problems}, series = {Mathematics of Control, Signals, and Systems}, journal = {Mathematics of Control, Signals, and Systems}, doi = {https://doi.org/10.1007/s00498-021-00280-4}, year = {2021}, abstract = {In this paper the turnpike phenomenon is studied for problems of optimal control where both pointwise-in-time state and control constraints can appear. We assume that in the objective function, a tracking term appears that is given as an integral over the time-interval [0, T] and measures the distance to a desired stationary state. In the optimal control problem, both the initial and the desired terminal state are prescribed. We assume that the system is exactly controllable in an abstract sense if the time horizon is long enough. We show that that the corresponding optimal control problems on the time intervals [0, T] give rise to a turnpike structure in the sense that for natural numbers n if T is su� ciently large, the contribution of the objective function from subintervals of [0, T] of the form [t - t/2^n, t + (T-t)/2^n] is of the order 1/min{t^n, (T-t)^n}. We also show that a similar result holds for epsilon-optimal solutions of the optimal control problems if epsilon > 0 is chosen suffciently 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.}, language = {en} } @unpublished{Gugat2021, author = {Gugat, Martin}, title = {Optimal boundary control of the wave equation: The finite-time turnpike phenomenon}, year = {2021}, abstract = {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 suffciently long. In optimal control problems, the desired terminal state is often enforced by terminal conditions, that add an additional diffculty 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 suffciently 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 in [1].}, language = {en} } @article{GugatUlbrich2017, author = {Gugat, Martin and Ulbrich, Stefan}, title = {On Lipschitz Solutions of Initial Boundary Value Problems for Balance Laws}, series = {Mathematical Models and Methods in Applied Sciences}, journal = {Mathematical Models and Methods in Applied Sciences}, year = {2017}, abstract = {The flow of gas through networks of pipes can be modeled by the isothermal Euler equations and algebraic node conditions that model the flow through the vertices of the network graph. We prove the well-posedness of the system for gas with nonconstant compressibility factor that is given by an affine linear function. We consider initial data and control functions that are Lipschitz continuous and compatible with the node and boundary conditions. We show the existence of semi--global Lipschitz continuous solutions of the initial boundary value problem. The construction of the solution is based upon a fixed point iteration along the characteristic curves. The solutions of the intial boundary value problem on arbitrary networks satisfy a maximum principle in terms of the Riemann invariants that states that the maximum of the absolute values is attained for the initial or the boundary data.}, language = {en} } @article{GugatLeugeringWang2017, author = {Gugat, Martin and Leugering, G{\"u}nter and Wang, Ke}, title = {Neumann boundary feedback stabilization for a nonlinear wave equation: A strict H2-Lyapunov function}, series = {Mathematical Control and Related Fields (MCRF)}, volume = {7}, journal = {Mathematical Control and Related Fields (MCRF)}, number = {3}, editor = {Zhang, Xu}, doi = {10.3934/mcrf.2017015}, pages = {419 -- 448}, year = {2017}, abstract = {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.}, language = {en} } @article{MartinHertyMueller2017, author = {Martin, Gugat and Herty, Michael and M{\"u}ller, Siegfried}, title = {Coupling conditions for the transition from supersonic to subsonic fluid states}, series = {Networks and Heterogeneous Media (NHM)}, volume = {12}, journal = {Networks and Heterogeneous Media (NHM)}, number = {3}, editor = {Piccoli, Benedetto}, doi = {10.3934/nhm.2017016}, pages = {371 -- 380}, year = {2017}, abstract = {We discuss coupling conditions for the p-system in case of a transition from supersonic states to subsonic states. A single junction with adjacent pipes is considered where on each pipe the gas ow is governed by a general p-system. By extending the notion of demand and supply known from traffic fiow analysis we obtain a constructive existence result of solutions compatible with the introduced conditions.}, language = {en} } @article{GugatZuazua2016, author = {Gugat, Martin and Zuazua, Enrique}, title = {Exact penalization of terminal constraints for optimal control problems}, series = {OPTIMAL CONTROL APPLICATIONS AND METHODS}, volume = {37}, journal = {OPTIMAL CONTROL APPLICATIONS AND METHODS}, number = {6}, editor = {Grimble, Mike J.}, doi = {10.1002/oca.2238}, pages = {1329 -- 1354}, year = {2016}, abstract = {We study optimal control problems for linear systems with prescribed initial and terminal states. We analyze the exact penalization of the terminal constraints. We show that for systems that are exactly controllable, the norm-minimal exact control can be computed as the solution of an optimization problem without terminal constraint but with a nonsmooth penalization of the end conditions in the objective function, if the penalty parameter is sufficiently large. We describe the application of the method for hyperbolic and parabolic systems of partial differential equations, considering the wave and heat equations as particular examples. Copyright © 2016 John Wiley \& Sons, Ltd.}, language = {en} } @article{GugatHante2017, author = {Gugat, Martin and Hante, Falk}, title = {Lipschitz Continuity of the Value Function in Mixed-Integer Optimal Control Problems}, series = {Mathematics of Control, Signals, and Systems}, volume = {29}, journal = {Mathematics of Control, Signals, and Systems}, number = {1}, doi = {10.1007/s00498-016-0183-4}, year = {2017}, language = {en} }