TY - JOUR A1 - Gugat, Martin A1 - Rosier, Lionel A1 - Perrollaz, Vincent ED - Arendt, W. ED - Pierre, M. T1 - Boundary stabilization of quasilinear hyperbolic systems of balance laws: exponential decay for small source terms JF - Journal of Evolution Equations N2 - We investigate the long-time behaviour of solutions of quasilinear hyperbolic systems with transparent boundary conditions when small source terms are incorporated in the system. Even if the finite-time stability of the system is not preserved, it is shown here that an exponential convergence towards the steady state still holds with a decay rate which is proportional to the logarithm of the amplitude of the source term. The result is stated for a system with dynamical boundary conditions in order to deal with initial data that are free of any compatibility condition. The proof of the existence and uniqueness of a solution defined for all positive times is also provided in this paper. Y1 - 2018 U6 - https://doi.org/10.1007/s00028-018-0449-z ER - TY - INPR A1 - Gugat, Martin A1 - Schuster, Michael A1 - Sokolowski, Jan T1 - Location Problem for Compressor Stations in Pipeline Networks 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 with the gas dynamics that governs the network flow. That results in non-convex 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, the problem of finding the optimal location for the control on the network, s.t. the control cost is minimal and the gas pressure stays within given bounds, is considered. In the deterministic setting, explicit bounds for the pipe length and the inlet pressure, s.t. a unique optimal compressor location with minimal control cost exists, are presented. In the probabilistic setting, an existence result for the optimal compressor location is presented and the uniqueness of the solution is discussed depending on the probability distribution. For Gaussian distributed loads a uniqueness result for the optimal compressor location is presented. Further the problem of finding the optimal compressor locations on networks including the number of compressor stations as variable is considered. 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 demonstrating that the compressor locations determined using a steady state approach are also admissible in transient settings. KW - Gas Networks KW - Compressor Control KW - Weber Problem KW - Optimal Location KW - Uncertain Boundary Data Y1 - 2024 ER - TY - JOUR A1 - Gugat, Martin A1 - Qian, Meizhi A1 - Sokolowski, Jan T1 - Network Design and Control: Shape and Topology Optimization for the Turnpike Property for the Wave Equation N2 - The optimal control problems for the wave equation are considered on networks. The turnpike property is shown for the state equation, the adjoint state equation as well as the optimal cost. The shape and topology optimization is performed for the network with the shape functional given by the optimality system of the control problem. The set of admissible shapes for the network is compact in finite dimensions, thus the use of turnpike property is straightforward. The topology optimization is analysed for an example of nucleation of a small cycle at the internal node of network. The topological derivative of the cost is introduced and evaluated in the framework of domain decomposition technique. Numerical examples are provided. KW - Turnpike Property KW - Wave Equation KW - Optimal Control KW - Spectral Method KW - Network Optimum Design Y1 - 2024 VL - J. Geom. Anal. IS - 34 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 T1 - New Lyapunov functions for systems with source terms JF - Control and Cybernetics N2 - Lyapunov functions with exponential weights have been used successfully as a powerful tool for the stability analysis of hyperbolic systems of balance laws. In this paper we extend the class of weight functions to a family of hyperbolic functions and study the advantages in the analysis of 2 × 2 systems of balance laws. We present cases connected with the study of the limit of stabilizability, where the new weights provide Lyapunov functions that show exponential stability for a larger set of problem parameters than classical exponential weights. Moreover, we show that sufficiently large time-delays influence the limit of stabilizability in the sense that the parameter set, for which the system can be stabilized becomes substantially smaller. We also demonstrate that the hyperbolic weights are useful in the analysis of the boundary feedback stability of systems of balance laws that are governed by quasilinear hyperbolic partial differential equations. KW - Lyapunov Function KW - Exponential Weight KW - Hyperbolic Weight KW - Feedback law KW - Stabilization Y1 - 2025 U6 - https://doi.org/https://doi.org/10.2478/candc-2024-0008 VL - 53 IS - 1 SP - 163 EP - 187 ER - TY - JOUR A1 - Gugat, Martin A1 - Ulbrich, Stefan T1 - On Lipschitz Solutions of Initial Boundary Value Problems for Balance Laws JF - Mathematical Models and Methods in Applied Sciences N2 - 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. Y1 - 2017 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 - Keimer, Alexander A1 - Leugering, Günter A1 - Wang, Zhiqiang ED - Piccoli, Benedetto T1 - Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks JF -  Networks and Heterogeneous Media N2 - We consider a system of scalar nonlocal conservation laws on networks that model a highly re-entrant multi-commodity manufacturing system as encountered in semiconductor production. Every single commodity is mod-eled by a nonlocal conservation law, and the corresponding PDEs are coupled via a collective load, the work in progress. We illustrate the dynamics for two commodities. In the applications, directed acyclic networks naturally occur, therefore this type of networks is considered. On every edge of the network we have a system of coupled conservation laws with nonlocal velocity. At the junctions the right hand side boundary data of the foregoing edges is passed as left hand side boundary data to the following edges and PDEs. For distributing junctions, where we have more than one outgoing edge, we impose time dependent distribution functions that guarantee conservation of mass. We provide results of regularity, existence and well-posedness of the multi-commodity network model for L p-, BV-and W 1,p-data. Moreover, we define an L 2-tracking type objective and show the existence of minimizers that solve the corresponding optimal control problem. KW - conservation laws on network KW - nonlocal conservation laws KW - optimal nodal control KW - systems of hyperbolic pdes Y1 - 2016 U6 - https://doi.org/DOI: 10.3934/nhm.2015.10.749 VL - 10 IS - 4 SP - 749 EP - 785 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 - Gugat, Martin A1 - Wintergerst, David A1 - Schultz, Rüdiger ED - Iske, Armin T1 - Networks of pipelines for gas with nonconstant compressibility factor: stationary states JF - Computational and Applied Mathematics N2 - For the management of gas transportation networks, it is essential to know how the stationary states of the system are determined by the boundary data. The isothermal Euler equations are an accurate pde-model for the gas flow through each pipe. A compressibility factor is used to model the nonlinear relationship between density and pressure that occurs in real gas in contrast to ideal gas. The gas flow through the nodes is governed by algebraic node conditions that require the conservation of mass and the continuity of the pressure. We examine networks that are described by arbitrary finite graphs and show that for suitably chosen boundary data, subsonic stationary states exist and are uniquely determined by the boundary data. Our construction of the stationary states is based upon explicit representations of the stationary states on each single pipe that can easily be evaluated numerically. We also use the monotonicity properties of these states as functions of the boundary data. Y1 - 2016 U6 - https://doi.org/10.1007/s40314-016-0383-z ER - TY - INPR A1 - Gugat, Martin A1 - Schultz, Rüdiger T1 - Boundary feedback stabilization of the isothermal Euler-equations with uncertain boundary data N2 - In a gas transport system, the customer behavior is uncertain. Motivated by this situation, we consider a boundary stabilization problem for the flow through a gas pipeline, where the outflow at one end of the pipe %that is governed by the customer's behavior is uncertain. The control action is located at the other end of the pipe. The feedback law is a classical Neumann velocity feedback with a feedback parameter $k>0$. 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 quasilinear 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. Y1 - 2017 ER - TY - CHAP A1 - Gugat, Martin A1 - Herty, Michael A1 - Yu, Hui ED - Westdickenberg, Michael T1 - On the relaxation approximation for 2 × 2 hyperbolic balance laws N2 - The relaxation approximation for systems of conservation laws has been studied intensively for example by [17, 5, 19, 24]. In this paper the corresponding relaxation approximation for 2x2 systems of balance laws is studied. Our driving example is gas flow in pipelines described by the isothermal Euler equations. We are interested in the limiting behavior as the relaxation parameter tends to zero. We give conditions where the relaxation converges to the states of the original system and counterexamples for cases where the steady states depend on the space variable. Y1 - 2017 VL - x IS - x SP - x EP - x PB - Springer CY - Springer Proceedings in Mathematics and Statistics ER - TY - JOUR A1 - Gugat, Martin A1 - Ulbrich, Stefan ED - CHEN, GOONG T1 - The isothermal Euler equations for ideal gas with source term: Product solutions, flow reversal and no blow up JF - Journal of Mathematical Analysis and Applications KW - Global classical solutions Ideal gas Bi-directional flow Transsonic flow Y1 - 2017 U6 - https://doi.org/10.1016/j.jmaa.2017.04.064 VL - 454 IS - 1 SP - 439 EP - 452 ER -