TY - JOUR A1 - Gugat, Martin A1 - Henrion, René A1 - Heitsch, Holger T1 - A turnpike property for optimal control problems with dynamic probabilistic constraints JF - Journal of Convex Analysis N2 - In this paper we consider systems that are governed by linear time-discrete dynamics with an initial condition and a terminal condition for the expected values. We study optimal control problems where in the objective function a term of tracking type for the expected values and a control cost appear. In addition, the feasible states have to satisfy a conservative probabilistic constraint that requires that the probability that the trajectories remain in a given set F is greater than or equal to a given lower bound. An application are optimal control problems related to storage management systems with uncertain in- and output. We give suffcient conditions that imply that the optimal expected trajectories remain close to a certain state that can be characterized as the solution of an optimal control problem without prescribed initial- and terminal condition. Hence we contribute to the study of the turnpike phenomenon that is well-known in mathematical economics. KW - Probabilistic Constraints KW - Probabilistic Robustness KW - here-and-now decision KW - Turnpike phenomenon KW - Measure turnpike Y1 - 2021 VL - 30 IS - 3 SP - 1025 EP - 1052 PB - Heldermann Verlag 2023 ER - TY - JOUR A1 - Gugat, Martin A1 - Weiland, Sven T1 - Nodal Stabilization of the Flow in a Network with a Cycle JF - Journal of Optimization, Differential Equations and their Applications N2 - In this paper we discuss an approach to the stability analysis for classical solutions of closed loop systems that is based upon the tracing of the evolution of the Riemann invariants along the characteristics. We consider a network where several edges are coupled through node conditions that govern the evolution of the Riemann invariants through the nodes of the network. The analysis of the decay of the Riemann invariants requires to follow backwards all the characteristics that enter such a node and contribute to the evolution. This means that with each nodal reflection/crossing the number of characteristics that contribute to the evolution increases. We show how for simple networks with a suffcient number of damping nodal controlers it is possible to keep track of this family of characteristics and use this approach to analyze the exponential stability of the system. The analysis is based on an adapted version of Gronwall's lemma that allows us to take into account the possible increase of the Riemann invariants when the characteristic curves cross a node of the network. Our example is motivated by applications in the control of gas pipeline flow, where the graphs of the networks often contain many cycles. KW - Nodal Stabilization KW - Classical Solutions KW - Cycled Networks KW - Gas Pipelines KW - Nodal Control Y1 - 2021 VL - 29 IS - 2 SP - 1 EP - 24 ER - TY - JOUR A1 - Giesselmann, Jan A1 - Gugat, Martin A1 - Kunkel, Teresa T1 - Observer-based data assimilation for barotropic gas transport using distributed measurements N2 - We consider a state estimation problem for gas pipeline flow modeled by the one-dimensional barotropic Euler equations. In order to reconstruct the system state, we construct an observer system of Luenberger type based on distributed measurements of one state variable. First, we show the existence of Lipschitz-continuous semi-global solutions of the observer system and of the original system for initial and boundary data satisfying smallness and compatibility conditions for a single pipe and for general networks. Second, based on an extension of the relative energy method we prove that the state of the observer system converges exponentially in the long time limit towards the original system state. We show this for a single pipe and for star-shaped networks. Y1 - 2023 U6 - https://doi.org/10.4310/CMS.240918203214 VL - 22 SP - 2271 EP - 2309 PB - Communications in Mathematical Sciences ER - TY - CHAP A1 - Gugat, Martin A1 - Schuster, Michael T1 - Max-p optimal boundary control of gas flow N2 - In the transition to renewable energy sources, hydrogen will potentially play an important role for energy storage. The efficient transport of this gas is possible via pipelines. An understanding of the possibilities to control the gas flow in pipelines is one of the main building blocks towards the optimal use of gas. For the operation of gas transport networks it is important to take into account the randomness of the consumers’ demand, where often information on the probability distribution is available. Hence in an efficient optimal control model the corresponding probability should be included and the optimal control should be such that the state that is generated by the optimal control satisfies given state constraints with large probability. We comment on the modelling of gas pipeline flow and the problems of optimal nodal control with random demand, where the aim of the optimization is to determine controls that generate states that satisfy given pressure bounds with large probability. We include the H2 norm of the control as control cost, since this avoids large pressure fluctuations which are harmful in the transport of hydrogen since they can cause embrittlement of the pipeline metal. KW - gas pipeline flow KW - nodal control KW - hyperbolic differential equation KW - random demand KW - state constraints Y1 - 2022 U6 - https://doi.org/https://doi.org/10.15495/EPub_UBT_00006809 VL - Extended Abstracts of the 25th International Symposium on Mathematical Theory of Networks and Systems Bayreuth, Germany, 12-16 September 2022 ER - TY - INPR A1 - Schuster, Michael A1 - Strauch, Elisa A1 - Wilka, Hendrik A1 - Lang, Jens A1 - Gugat, Martin T1 - Probabilistic Robustness for Compressor Controls in Transient Pipeline Networks N2 - Uncertainty plays a crucial role in modeling and optimization of complex systems across various applications. In this paper, uncertain gas transport through pipeline networks is considered and a novel strategy to measure the robustness of deterministically computed compressor and valve controls, the probabilistic robustness, is presented. \noindent Initially, an optimal control for a deterministic gas network problem is computed such that the total control cost is minimized with respect to box constraints for the pressure. Subsequently, the model is perturbed by uncertain gas demands. The probability, that the uncertain gas pressures - based on the a priori deterministic optimal control - satisfy the box constraints, is evaluated. Moreover, buffer zones are introduced in order to tighten the pressure bounds in the deterministic scenario. Optimal controls for the deterministic scenario with buffer zones are also applied to the uncertain scenario, allowing to analyze the impact of the buffer zones on the probabilistic robustness of the optimal controls. \noindent For the computation of the probability, we apply a kernel density estimator based on samples of the uncertain pressure at chosen locations. In order to reduce the computational effort of generating the samples, we combine the kernel density estimator approach with a stochastic collocation method which approximates the pressure at the chosen locations in the stochastic space. Finally, we discuss generalizations of the probabilistic robustness check and we present numerical results for a gas network taken from the public gas library. KW - Probabilistic Robustness KW - Gas Network Control KW - Probabilistic Constrained Optimization KW - Stochastic Collocation KW - Kernel Density Estimation Y1 - 2025 ER - TY - INPR A1 - Breitkopf, Jannik A1 - Gugat, Martin A1 - Ulbrich, Stefan T1 - Existence and Optimal Boundary Control of Classical Solutions to Networks of Quasilinear Hyperbolic Systems of Balance Laws N2 - We study the existence, stability and optimal control of classical solutions of networked strictly hyperbolic systems of balance laws. Such networks arise, for instance, in the modeling of the gas dynamics in a network of gas pipelines, traffic networks, or networks of water channels. It is assumed that characteristic speeds are nonzero and do not change sign. Node conditions are stated in a general way by requiring that the boundary traces of all states adjacent to a vertex satisfy an algebraic equation. With a suitable assumption, this equation can be solved for outgoing characteristic variables as a function of the incoming ones. We prove the unique existence of a classical solution for arbitrarily large initial and boundary values on a possibly small time horizon. We derive continuity and differentiability properties of the solution operator of the quasilinear problem w.r.t. the control term located in the node conditions. Then we analyze an optimal control problem for the networked system, where the control is of boundary type. We prove the existence of optimal controls and the differentiability of the objective functional w.r.t. the controls. KW - networked systems, classical solutions, quasilinear hyperbolic systems, boundary control, conservation laws, nodal control, optimal nodal control Y1 - 2025 ER - TY - JOUR A1 - Gugat, Martin T1 - Boundary Stabilization of Quasi-Linear Hyperbolic Systems with Varying Time-Delay N2 - In this paper we consider the boundary feedback stabilization of a quasi-linear hyperbolic system of balance laws. At one end of the space interval, there is a reflecting boundary condition. At the other end a stabilizing feedback law with a varying time-delay is prescribed. We present sufficient conditions for the exponential stability of the system. We show that exponential stabilization is possible if the product of the length of the interval and an upper bound for the source term is sufficiently small. We also show that if the product of the length of the interval and a lower bound for the source term is sufficiently large, the system is unstable. Our analysis is based on Lyapunov functions with weights that are given by hyperbolic functions that generalize the well-known exponential weights. Compared with previous contributions, we obtain conditions that can be verified more easily in terms of the system parameters. Our results show that for sufficiently short space intervals, and also with varying time-delay, exponential stabilization is possible with appropriately chosen feedback gains that depend on the maximal value of the time-delay and the maximal absolute value of its derivative. KW - boundary stabilization KW - feedback law KW - varying delay KW - quasi-linear hyperbolic system KW - boundary control Y1 - 2026 U6 - https://doi.org/10.1137/24M1648570 VL - 63 IS - SIAM Journal on Control and Optimization SP - 452 EP - 471 ER - TY - JOUR A1 - Gugat, Martin A1 - Schuster, Michael T1 - Optimal Neumann Control of the Wave Equation with L1-Control Cost: The Finite-Time Turnpike Property JF - Optimization N2 - The finite-time turnpike property describes a situation where the optimal state reaches a steady state after finite time. The steady state is a solution of a static optimal control problem. We study an optimal control problem where the objective functional is the sum of an 𝐿1-norm control cost with a weight 𝛾>0 and a differentiable tracking term. We consider a vibrating string with homogeneous Dirichlet conditions at one end and Neumann control action at the other end. The tracking term is defined by the squared 𝐿2-norm of a non-collocated Neumann-observation. We show that the problem has a unique solution and that due to the non-smoothness of the 𝐿1-norm for sufficiently large T the optimal state reaches the desired state after a finite time that is equal to the minimal time where exact controllability holds. We also study the effect of smoothing of the control cost on the structure of the optimal control and show that the finite-time turnpike property also holds in the smoothing limit in a strong 𝐿2-sense. Y1 - 2024 U6 - https://doi.org/https://doi.org/10.1080/02331934.2024.2373904 ER - TY - JOUR A1 - Gugat, Martin T1 - Synchronization of velocities in pipeline flow of blended gas JF - Journal of Mathematical Analysis and Applications N2 - We consider the pipeline flflow of blended gas. The flow is governed by a coupled system where for each component we have the isothermal Euler equations with an additional velocity coupling term that couples the velocities of the different components. Our motivation is hydrogen blending in natural gas pipelines, which will play a role in the transition to renewable energies. We show that with suitable boundary conditions the velocities of the gas components synchronize exponentially fast, as long as the L2-norm of the synchronization error is outside of a certain interval where the size of the interval is determined by the order of the interaction terms. This indicates that in some cases for a mixture of ncomponents it is justifified to use a flux model where it is assumed that all components flow with the same velocity. For the proofs we use an appropriately chosen Lyapunov function which is based upon the idea of relative energy. KW - Synchronization of solutions to PDEs KW - Quasi-linear hyperbolic PDE KW - Drift-flux model KW - Lyapunov function KW - Relative energy Y1 - 2026 U6 - https://doi.org/10.1016/j.jmaa.2025.130078 VL - 556 IS - 1 ER - TY - JOUR A1 - Gugat, Martin A1 - Giesselmann, Jan A1 - Kunkel, Teresa T1 - Exponential synchronization of a nodal observer for a semilinear model for the flow in gas networks N2 - 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. Y1 - 2021 U6 - https://doi.org/10.1093/imamci/dnab029 CY - IMA Journal of Mathematical Control and Information ER - TY - JOUR A1 - Gugat, Martin A1 - Herty, Michael A1 - Müller, Siegfried ED - Piccoli, Benedetto T1 - Coupling conditions for the transition from supersonic to subsonic fluid states JF - Networks and Heterogeneous Media (NHM) N2 - 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. KW - p-system KW - coupling conditions KW - network Y1 - 2017 U6 - https://doi.org/10.3934/nhm.2017016 VL - 12 IS - 3 SP - 371 EP - 380 ER -