TY - INPR A1 - Göttlich, Simone A1 - Schuster, Michael A1 - Ulke, Alena T1 - On the Existence of Steady States for Blended Gas Flow with Non-Constant Compressibility Factor on Networks N2 - In this paper, we study hydrogen-natural gas mixtures transported through pipeline networks. The flow is modeled by the isothermal Euler equations with a pressure law involving a non-constant, composition-dependent compressibility factor. For a broad class of such compressibility models, we prove the existence of steady-state solutions on networks containing compressor stations. The analysis is based on an implicit representation of the pressure profiles and a continuity argument that overcomes the discontinuous dependence of the gas composition on the flow direction. Numerical examples illustrate the influence of different compressibility models on the resulting states. KW - Blended Gas Flow KW - Compressibility Factor KW - z-Factor KW - Real Gas KW - Steady States Y1 - 2026 ER - TY - JOUR A1 - Heitsch, Holger A1 - Henrion, René T1 - On the Lipschitz continuity of the spherical cap discrepancy around generic point sets JF - Unif. Distrib. Theory N2 - The spherical cap discrepancy is a prominent measure of uniformity for sets on the d-dimensional sphere. It is particularly important for estimating the integration error for certain classes of functions on the sphere. Building on a recently proven explicit formula for the spherical discrepancy, we show as a main result of this paper that this discrepancy is Lipschitz continuous in a neighbourhood of so-called generic point sets (as they are typical outcomes of Monte-Carlo sampling). This property may have some impact (both algorithmically and theoretically for deriving necessary optimality conditions) on optimal quantization, i.e., on finding point sets of fixed size on the sphere having minimum spherical discrepancy. KW - spherical cap discrepancy KW - uniform distribution on sphere KW - Lipschitz continuity KW - necessary optimality conditions Y1 - 2025 U6 - https://doi.org/10.2478/udt-2025-0011 VL - 20 IS - 1 SP - 35 EP - 63 ER - TY - JOUR A1 - Bernhard, Daniela A1 - Heitsch, Holger A1 - Henrion, René A1 - Liers, Frauke A1 - Stingl, Michael A1 - Uihlein, Andrian A1 - Zipf, Viktor T1 - Continuous stochastic gradient and spherical radial decomposition N2 - In this paper, a new method is presented for solving chance-constrained optimization problems. The method combines the well-established Spherical-Radial Decomposition approach with the Continuous Stochastic Gradient method. While the Continuous Stochastic Gradient method has been successfully applied to chance-constrained problems in the past, only the combination with the Spherical-Radial Decomposition allows to avoid smoothing of the integrand. In this chapter, we prove this fact for a relevant class of chance-constrained problems and apply the resulting method to the capacity maximization problem for gas networks. KW - chance constraints KW - continuous stochastic gradient KW - spheric-radial decomposition Y1 - ER - TY - THES A1 - Schuster, Michael T1 - Control and Optimization under Uncertainty in the Context of Gas Network Operation N2 - The transition to renewable energy and the increasing role of hydrogen as a future energy carrier pose major challenges for the operation and optimization of gas transport networks. A rigorous mathematical understanding of the topic is necessary as efficient and reliable operation requires advanced methods to deal with uncertainty, nonlinear dynamics, and complex network topologies. At the same time, optimal control theory – in particular the turnpike phenomenon – provides powerful tools to simplify long-term optimization problems and to connect dynamic models with their stationary counterparts. This habilitation thesis focuses on establishing new results in three related areas. For optimization problems with probabilistic constraints, an approach for approximating probabilities based on kernel density estimation is presented and analyzed, yielding necessary and sufficient conditions for the convergence of solutions of approximated stochastic optimization problems. In the modeling of gas transport networks, the existence and uniqueness results of a mixing model are presented. Furthermore, a finite-time turnpike result for an optimal control problem governed by the wave equation as well as an integral turnpike result for an optimal control problem governed by the transport equation under uncertainty is established. On the application side, the problem of optimal placement of compressor stations in stationary gas networks is analyzed for both deterministic and random gas demand. Results on the optimal number of compressor stations and their locations on gas networks are presented. Further, the turnpike phenomenon is applied to an optimal control and design problem for gas networks, allowing steady-state models to replace the gas dynamics in the long-term planning. For the corresponding stationary problem, a fast and efficient algorithm for identifying the optimal network topology with low control cost is provided. N2 - Die Energiewende und die zunehmende Bedeutung von Wasserstoff als zukünftiger Energieträger stellen erhebliche Herausforderungen an den Betrieb und die Optimierung von Gastransportnetzen. Ein detailliertes, mathematisches Verständnis ist notwendig, da ein effizienter und sicherer Betrieb komplexe Methoden im Umgang mit Unsicherheiten, nichtlinearen Transportdynamiken und komplexen Netzstrukturen erfordert. Gleichzeitig liefert die Optimalsteuerungstheorie – insbesondere das Turnpike-Phänomen – wirkungsvolle Werkzeuge, um langzeitorientierte Optimierungsprobleme zu vereinfachen und dynamische Modelle mit ihren stationären Gegenstücken zu verbinden. Diese Habilitationsschrift konzentriert sich auf die Entwicklung neuer Ergebnisse in drei verwandten Forschungsbereichen. Für Optimierungsprobleme mit probabilistischen Nebenbedingungen wird ein Ansatz zur Approximation von Wahrscheinlichkeiten basierend auf Kerndichteschätzern präsentiert und analysiert, der notwendige und hinreichende Bedingungen für die Konvergenz von Lösungen approximierter stochastischer Optimierungsprobleme liefert. In der Modellierung von Gastransportnetzen werden Existenz- und Eindeutigkeitsresultate für ein Mischungsmodell hergeleitet. Darüber hinaus wird ein Finite-Time-Turnpike-Ergebnis für ein Optimalsteuerungsproblem mit der Wellengleichung sowie ein Integral-Turnpike-Ergebnis für ein Optimalsteuerungsproblem mit Transportgleichungen unter Unsicherheit präsentiert. Auf der Anwendungsseite wird das Problem der optimalen Platzierung von Verdichterstationen in stationären Gasnetzen sowohl für deterministische als auch für stochastische Gasnachfrage untersucht. Ergebnisse zur optimalen Anzahl und Positionierung von Verdichtern werden vorgestellt. Ferner wird das Turnpike-Phänomen auf ein kombiniertes Steuerungs- und Designproblem für Gasnetze angewandt, wodurch stationäre Modelle anstelle der Gasdynamik in der langfristigen Planung genutzt werden können. Für das entsprechende stationäre Problem wird ein schnelles und effizientes Verfahren zur Bestimmung optimaler Netzwerktopologien mit geringen Steuerungskosten bereitgestellt. Y1 - 2025 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 - CHAP A1 - Pfetsch, Marc A1 - Schmidt, Martin A1 - Skutella, Martin A1 - Thürauf, Johannes T1 - Potential-Based Flows - An Overview T2 - Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks N2 - Potential-based flows provide an algebraic way to model static physical flows in networks, for example, in gas, water, and lossless DC power networks. The flow on an arc in the network depends on the difference of the potentials at its end-nodes, possibly in a nonlinear way. Potential-based flows have several nice properties like uniqueness and acyclicity. The goal of this paper is to provide an overview of the current knowledge on these models with a focus on optimization problems on such networks. We cover basic properties, computational complexity, monotonicity, uncertain parameters, and the corresponding behavior of the network as well as topology optimization. Y1 - 2026 PB - Springer-Nature 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 - INPR A1 - Giesselmann, Jan A1 - Ranocha, Hendrik T1 - Convergence of hyperbolic approximations to higher-order PDEs for smooth solutions N2 - We prove the convergence of hyperbolic approximations for several classes of higher-order PDEs, including the Benjamin-Bona-Mahony, Korteweg-de Vries, Gardner, Kawahara, and Kuramoto-Sivashinsky equations, provided a smooth solution of the limiting problem exists. We only require weak (entropy) solutions of the hyperbolic approximations. Thereby, we provide a solid foundation for these approximations, which have been used in the literature without rigorous convergence analysis. We also present numerical results that support our theoretical findings. 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 - INPR A1 - Bernhard, Daniela A1 - Liers, Frauke A1 - Stingl, Michael T1 - Branch-and-cut for mixed-integer robust chance-constrained optimization with discrete distributions N2 - We study robust chance-constrained problems with mixed-integer design variables and ambiguity sets consisting of discrete probability distributions. Allowing general non-convex constraint functions, we develop a branch-and-cut framework using scenario-based cutting planes to generate lower bounds. The cutting planes are obtained by exploiting the classical big-M reformulation of the chance-constrained problem in the case of discrete distributions. Furthermore, we include the calculation of initial feasible solutions based on a bundle method applied to an approximation of the original problem into the branch-and-cut procedure. We conclude with a detailed discussion about the practical performance of the branch-and-cut framework with and without initial feasible solutions. In our experiments we focus on gas transport problems under uncertainty and provide a comparison of our method with solving the classical reformulation directly for various real-world sized instances. Y1 - ER - TY - INPR A1 - Breitkopf, Jannik A1 - Ulbrich, Stefan T1 - A Variational Calculus for Optimal Control of the Generalized Riemann Problem for Hyperbolic Systems of Conservation Laws N2 - We develop a variational calculus for entropy solutions of the Generalized Riemann Problem (GRP) for strictly hyperbolic systems of conservation laws where the control is the initial state. The GRP has a discontinuous initial state with exactly one discontinuity and continuously differentiable (C^1) states left and right of it. The control consists of the C^1 parts of the initial state and the position of the discontinuity. Solutions of the problem are generally discontinuous since they contain shock curves. We assume the time horizon T>0 to be sufficiently small such that no shocks interact and no new shocks are generated. Moreover, we assume that no rarefaction waves occur and that the jump of the initial state is sufficiently small. Since the shock positions depend on the control, a transformation to a reference space is used to fix the shock positions. In the reference space, we prove that the solution of the GRP between the shocks is continuously differentiable from the control space to C^0. In physical coordinates, this implies that the shock curves in C^1 and the states between the shocks in the topology of C^0 depend continuously differentiable on the control. As a consequence, we obtain the differentiability of tracking type objective functionals. KW - hyperbolic systems of conservation laws, shock curves, generalized riemann problem, optimal control, variational calculus Y1 - 2025 ER - TY - INPR A1 - Denzler, Sebastian A1 - Aigner, Kevin-Martin A1 - Lüer, Larry A1 - Brabec, Christoph A1 - Liers, Frauke T1 - Robust Bayesian Optimization with an Application to Material Science N2 - We propose a novel online learning framework for robust Bayesian optimization of uncertain black-box functions. While Bayesian optimization is well-suited for data-efficient optimization of expensive objectives, its standard form can be sensitive to hidden or varying parameters. To address this issue, we consider a min–max robust counterpart of the optimization problem and develop a practically efficient solution algorithm, BROVER (Bayesian Robust Optimization via Exploration with Regret minimization). Our method combines Gaussian process regression with a decomposition approach: the minimax structure is split into a non-convex online learner based on the Follow-the-Perturbed-Leader algorithm together with a subsequent minimization step in the decision variables. We prove that the theoretical regret bound converges under mild assumptions, ensuring asymptotic convergence to robust solutions. Numerical experiments on synthetic data validate the regret guarantees and demonstrate fast convergence to the robust optimum. Furthermore, we apply our method to the robust optimization of organic solar cell performance, where hidden process parameters and experimental variability naturally induce uncertainty. Our results on real-world datae show that BROVER identifies solutions with strong robustness properties within relatively few iterations, thereby offering a modern and practical approach for data-driven black-box optimization under uncertainty. KW - robust optimization KW - Bayesian optimization KW - online learning KW - solar cell performance Y1 - 2025 ER - TY - INPR A1 - Ulke, Alena A1 - Schuster, Michael A1 - Göttlich, Simone T1 - Steady State Blended Gas Flow on Networks: Existence and Uniqueness of Solutions N2 - We prove an existence result for the steady state flow of gas mixtures on networks. The basis of the model are the physical principles of the isothermal Euler equation, coupling conditions for the flow and pressure, and the mixing of incoming flow at nodes. The state equation is based on a convex combination of the ideal gas equations of state for natural gas and hydrogen. We analyze mathematical properties of the model allowing us to prove the existence of solutions in particular for tree-shaped networks and networks with exactly one cycle. Numerical examples illustrate the results and explore the applicability of our approach to different network topologies. KW - gas pipeline network KW - gas mixture KW - hydrogen blending KW - isothermal Euler equations KW - stationary states Y1 - 2024 VL - Netw. Heterog. Media ER - TY - INPR A1 - Schuster, Michael A1 - Sokolowski, Jan T1 - The Topological Derivative Method for Optimum Shape Design and Control of Gas Networks N2 - In this paper, topological derivatives are defined and employed for gas transport networks governed by nonlinear hyperbolic systems of PDEs. The concept of topological derivatives of a shape functional is introduced for optimum design and control of gas networks. First, the dynamic model for the network is considered. The cost for the control problem includes the deviations of the pressure at the inflow and outflow nodes. For dynamic control problems of gas networks when the turnpike property occurs, the synthesis of control and optimum design of the network can be simplified. That is, the design of the network can be performed for optimal control of the steady-state network model. The cost of design is defined by the optimal control cost for the steady-state network model. The topological derivative of the design cost, given by the optimal control cost with respect to the nucleation of a small cycle, is determined. Tree-structured networks can be decomposed into single network junctions. The topological derivative of the design cost is systematically evaluated at each junction of the decomposed network. This allows for the identification of internal nodes with negative topological derivatives, where replacing the node with a small cycle leads to an improved design cost. As the set of network junctions is finite, the iterative procedure is convergent. This design procedure is applied to representative examples and it can be generalized to arbitrary network graphs. A key feature of such modeling approach is the availability of exact steady-state solutions, enabling a fully analytical topological analysis of the design cost without numerical approximations. KW - Gas Networks KW - Optimum Design KW - Optimal Control KW - Topological Derivative KW - Turnpike Phenomenon Y1 - 2025 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 - Bernhard, Daniela A1 - Stingl, Michael A1 - Liers, Frauke T1 - Algorithms for robust chance-constrained optimization with mixture ambiguity N2 - Constructing ambiguity sets in distributionally robust optimization is difficult and currently receives increased attention. In this paper, we focus on mixture models with finitely many reference distributions. We present two different solution concepts for robust joint chance-constrained optimization problems with these ambiguity sets and non-convex constraint functions. Both concepts rely on solving an approximation problem that is based on well-known smoothing and penalization techniques. On the one side, we consider a classical bundle method together with an approach for finding good starting points. On the other side, we integrate the Continuous Stochastic Gradient method, a variant of the stochastic gradient descent that is able to exploit regularity in the data. On the example of gas networks we compare the two algorithmic concepts for different topologies and two types of mixture ambiguity sets with Gaussian reference distributions and polyhedral and ϕ-divergence based feasible sets for the mixing coefficients. The results show that both solution approaches are well-suited to solve this difficult problem class. Based on the numerical results we provide some general advices for choosing the more efficient algorithm depending on the main challenges of the considered optimization problem. We give an outlook for the applicability of the method in a wider context. Y1 - ER - TY - CHAP A1 - Disser, Yann A1 - Griesbach, Svenja M. A1 - Klimm, Max A1 - Lutz, Annette T1 - Bicriterial Approximation for the Incremental Prize-Collecting Steiner-Tree Problem N2 - We consider an incremental variant of the rooted prize-collecting Steiner-tree problem with a growing budget constraint. While no incremental solution exists that simultaneously approximates the optimum for all budgets, we show that a bicriterial (α,μ)-approximation is possible, i.e., a solution that with budget B+α for all B∈R≥0 is a multiplicative μ-approximation compared to the optimum solution with budget B. For the case that the underlying graph is a tree, we present a polynomial-time density-greedy algorithm that computes a (χ,1)-approximation, where χ denotes the eccentricity of the root vertex in the underlying graph, and show that this is best possible. An adaptation of the density-greedy algorithm for general graphs is (γ,2)-competitive where γ is the maximal length of a vertex-disjoint path starting in the root. While this algorithm does not run in polynomial time, it can be adapted to a (γ,3)-competitive algorithm that runs in polynomial time. We further devise a capacity-scaling algorithm that guarantees a (3χ,8)-approximation and, more generally, a ((4ℓ−1)χ,(2^(ℓ+2))/(2^ℓ−1))-approximation for every fixed ℓ∈N. KW - incremental maximization KW - competitive analysis KW - prize-collecting Steiner-tree Y1 - 2024 ER - TY - INPR A1 - Berrens, Arne A1 - Giesselmann, Jan T1 - A posteriori error control for a finite volume scheme for a cross-diffusion model of ion transport N2 - We derive a reliable a posteriori error estimate for a cell-centered finite volume scheme approximating a cross-diffusion system modeling ion transport through nanopores. To this end we derive an abstract stability framework that is independent of the numerical scheme and introduce a suitable (conforming) reconstruction of the numerical solution. The stability framework relies on some simplifying assumption that coincide with those made in weak uniqueness results for this system. This is the first a posteriori error estimate for a cross-diffusion system. Along the way, we derive a pointwise a posteriori error estimate for a finite volume scheme approximating the diffusion equation. We conduct numerical experiments showing that the error estimator scales with the same order as the true error. KW - cross-diffusion KW - ion transport KW - finite-volume approximation KW - a posteriori error estimates KW - diffusion equation Y1 - 2025 ER - TY - INPR A1 - Birke, Gunnar A1 - Engwer, Christian A1 - Giesselmann, Jan A1 - May, Sandra T1 - Error analysis of a first-order DoD cut cell method for 2D unsteady advection N2 - In this work we present an a priori error analysis for solving the unsteady advection equation on cut cell meshes along a straight ramp in two dimensions. The space discretization uses a lowest order upwind-type discontinuous Galerkin scheme involving a \textit{Domain of Dependence} (DoD) stabilization to correct the update in the neighborhood of small cut cells. Thereby, it is possible to employ explicit time stepping schemes with a time step length that is independent of the size of the very small cut cells. Our error analysis is based on a general framework for error estimates for first-order linear partial differential equations that relies on consistency, boundedness, and discrete dissipation of the discrete bilinear form. We prove these properties for the space discretization involving DoD stabilization. This allows us to prove, for the fully discrete scheme, a quasi-optimal error estimate of order one half in a norm that combines the L∞-in-time L2-in-space norm and a seminorm that contains velocity weighted jumps. We also provide corresponding numerical results. KW - cut cell KW - discontinuous Galerkin method KW - DoD Stabilization KW - a priori error estimate KW - unsteady advection Y1 - 2024 ER - TY - INPR A1 - Giesselmann, Jan A1 - Kwon, Kiwoong A1 - Lee, Min-Gi T1 - Relative entropy technique in terms of position and momentum and its application to Euler-Poisson system N2 - This paper presents a systematic study of the relative entropy technique for compressible motions of continuum bodies described as Hamiltonian flows. While the description for the classical mechanics of N particles involves a Hamiltonian in terms of position and momentum vectors, that for the continuum fluid involves a Hamiltonian in terms of density and momentum. For space dimension d≥2, the Hamiltonian functional has a non-convex dependency on the deformation gradient or placement map due to material frame indifference. Because of this, the applicability of the relative entropy technique with respect to the deformation gradient or the placement map is inherently limited. Despite these limitations, we delineate the feasible applications and limitations of the technique by pushing it to its available extent. Specifically, we derive the relative Hamiltonian identity, where the Hamiltonian takes the position and momentum field as its primary and conjugate state variables, all within the context of the referential coordinate system that describes the motion. This approach, when applicable, turns out to yield rather strong stability statements. As instances, we consider Euler-Poisson systems in one space dimension. For a specific pressureless model, we verify non-increasing L2 state differences before the formation of δ-shock. In addition, weak-strong uniqueness, stability of rarefaction waves, and convergence to the gradient flow in the singular limit of large friction are shown. Depending on the presence or absence of pressure, assumptions are made to suitably accommodate phenomena such as δ-shocks, vacuums, and shock discontinuities in the weak solutions. Y1 - 2024 ER - TY - INPR A1 - Topalovic, Antonia A1 - Hante, Falk M. T1 - Stabilizing Model Predictive Control for Generalized Nash Equilibrium Problems using Approximation by α -quasi-GNEPs and Lyapunov End Cost N2 - We study model predictive control (MPC) schemes for non-cooperative dynamic games regarding stabilization. The dynamic games are modelled as generalized Nash equilibrium problems (GNEPs), in which a shared constraint is given as a jointly controlled time-discrete (linear) dynamics. Furthermore, the players’ objectives are interdependent. We present recent results concerning their stabilizing properties using α-quasi- GENP-approximation and terminal conditions in the form of equilibrium endpoint constraints. Moreover, we extend the result towards Lyapunov terminal costs, which is a more general type of terminal condition. Furthermore, we show that a suitable Lyapunov terminal cost can be obtained from a non-game- based MPC scheme. This non-game-based MPC scheme relies on a classical optimal control problem for the aggregated cost. Hence, known results for determining the Lyapunov cost can be applied and carried over to the game-based setting. The theoretical results are complemented by numerical experiments. Y1 - 2024 ER - TY - INPR A1 - Aigner, Kevin-Martin A1 - Goerigk, Marc A1 - Hartisch, Michael A1 - Liers, Frauke A1 - Miehlich, Arthur A1 - Rösel, Florian T1 - Feature Selection for Data-Driven Explainable Optimization N2 - Mathematical optimization, although often leading to NP-hard models, is now capable of solving even large-scale instances within reasonable time. However, the primary focus is often placed solely on optimality. This implies that while obtained solutions are globally optimal, they are frequently not comprehensible to humans, in particular when obtained by black-box routines. In contrast, explainability is a standard requirement for results in Artificial Intelligence, but it is rarely considered in optimization yet. There are only a few studies that aim to find solutions that are both of high quality and explainable. In recent work, explainability for optimization was defined in a data-driven manner: a solution is considered explainable if it closely resembles solutions that have been used in the past under similar circumstances. To this end, it is crucial to identify a preferably small subset of features from a presumably large set that can be used to explain a solution. In mathematical optimization, feature selection has received little attention yet. In this work, we formally define the feature selection problem for explainable optimization and prove that its decision version is NP-complete. We introduce mathematical models for optimized feature selection. As their global solution requires significant computation time with modern mixed-integer linear solvers, we employ local heuristics. Our computational study using data that reflect real-world scenarios demonstrates that the problem can be solved practically efficiently for instances of reasonable size. KW - Feature selection KW - Explainable optimization KW - Data-driven optimization Y1 - 2025 ER - TY - INPR A1 - Hildebrand, Robert A1 - Göß, Adrian T1 - Complexity of Integer Programming in Reverse Convex Sets via Boundary Hyperplane Cover N2 - We study the complexity of identifying the integer feasibility of reverse convex sets. We present various settings where the complexity can be either NP-Hard or efficiently solvable when the dimension is fixed. Of particular interest is the case of bounded reverse convex constraints with a polyhedral domain. We introduce a structure, Boundary Hyperplane Cover, that permits this problem to be solved in polynomial time in fixed dimension provided the number of nonlinear reverse convex sets is fixed. Y1 - ER - TY - INPR A1 - Göß, Adrian A1 - Burlacu, Robert A1 - Martin, Alexander T1 - Parabolic Approximation & Relaxation for MINLP N2 - We propose an approach based on quadratic approximations for solving general Mixed-Integer Nonlinear Programming (MINLP) problems. Specifically, our approach entails the global approximation of the epigraphs of constraint functions by means of paraboloids, which are polynomials of degree two with univariate quadratic terms, and relies on a Lipschitz property only. These approximations are then integrated into the original problem. To this end, we introduce a novel approach to compute globally valid epigraph approximations by paraboloids via a Mixed-Integer Linear Programming (MIP) model. We emphasize the possibility of performing such approximations a-priori and providing them in form of a lookup table, and then present several ways of leveraging the approximations to tackle the original problem. We provide the necessary theoretical background and conduct computational experiments on instances of the MINLPLib. As a result, this approach significantly accelerates the solution process of MINLP problems, particularly those involving many trigonometric or few exponential functions. In general, we highlight that the proposed technique is able to exploit advances in Mixed-Integer Quadratically-Constrained Programming (MIQCP) to solve MINLP problems. KW - mixed-integer nonlinear programming KW - mixed-integer linear programming KW - quadratic approximation KW - global optimization Y1 - 2025 ER - TY - RPRT A1 - Amer, Zeina A1 - Avdzhieva, Ana A1 - Bongarti, Marcelo A1 - Dvurechensky, Pavel A1 - Farrell, Patricio A1 - Gotzes, Uwe A1 - Hante, Falk M. A1 - Karsai, Attila A1 - Kater, Stefan A1 - Liero, Matthias A1 - Spreckelsen, Klaus A1 - Taraz, Johannes A1 - Peschka, Dirk A1 - Plato, Luisa T1 - Modeling Hydrogen Embrittlement for Pricing Degradation in Gas Pipelines N2 - This paper addresses the critical challenge of hydrogen embrittlement in the context of Germany’s transition to a sustainable, hydrogen-inclusive energy system. As hydrogen infrastructure expands, estimating and pricing embrittlement become paramount due to safety, operational, and economic concerns. We present a twofold contribution: (1) We discuss hydrogen embrittlement modeling using both continuum models and simplified approximations. (2) Based on these models, we propose optimization-based pricing schemes for market makers, considering simplified cyclic loading and more complex digital twin models. Our approaches leverage widely-used subcritical crack growth models in steel pipelines, with parameters derived from experiments. The study highlights the challenges and potential solutions for incorporating hydrogen embrittlement into gas transportation planning and pricing, ultimately aiming to enhance the safety and economic viability of Germany’s future energy infrastructure. Y1 - 2025 ER - TY - JOUR A1 - Giesselmann, Jan A1 - Karsai, Attila A1 - Tscherpel, Tabea T1 - Energy-consistent Petrov-Galerkin time discretization of port-Hamiltonian systems N2 - For a general class of nonlinear port-Hamiltonian systems we develop a high-order time discretization scheme with certain structure preservation properties. The finite or infinite-dimensional system under consideration possesses a Hamiltonian function, which represents an energy in the system and is conserved or dissipated along solutions. For infinite-dimensional systems this structure is preserved under suitable Galerkin discretization in space. The numerical scheme is energy-consistent in the sense that the Hamiltonian of the approximate solutions at time grid points behaves accordingly. This structure preservation property is achieved by specific design of a continuous Petrov-Galerkin (cPG) method in time. It coincides with standard cPG methods in special cases, in which the latter are energy-consistent. Examples of port-Hamiltonian ODEs and PDEs are presented to visualize the framework. In numerical experiments the energy consistency is verified and the convergence behavior is investigated. Y1 - 2025 U6 - https://doi.org/10.48550/arXiv.2404.12480 ER - TY - JOUR A1 - Breiten, Tobias A1 - Karsai, Attila T1 - Structure-preserving $H_\infty$ control for port-Hamiltonian systems N2 - We study $H_\infty$ control design for linear time-invariant port-Hamiltonian systems. By a modification of the two central algebraic Riccati equations, we ensure that the resulting controller will be port-Hamiltonian. Using these modified equations, we proceed to show that a corresponding balanced truncation approach preserves port-Hamiltonian structure. We illustrate the theoretical findings using numerical examples and observe that the chosen representation of the port-Hamiltonian system can have an influence on the approximation qualities of the reduced order model. Y1 - 2025 U6 - https://doi.org/10.1016/j.sysconle.2023.105493 ER - TY - JOUR A1 - Karsai, Attila T1 - Manifold turnpikes of nonlinear port-Hamiltonian descriptor systems under minimal energy supply N2 - Turnpike phenomena of nonlinear port-Hamiltonian descriptor systems under minimal energy supply are studied. Under assumptions on the smoothness of the system nonlinearities, it is shown that the optimal control problem is dissipative with respect to a manifold. Then, under controllability assumptions, it is shown that the optimal control problem exhibits a manifold turnpike property. Y1 - 2025 U6 - https://doi.org/10.1007/s00498-024-00384-7 ER - TY - JOUR A1 - Karsai, Attila A1 - Breiten, Tobias A1 - Ramme, Justus A1 - Schulze, Philipp T1 - Nonlinear port-Hamiltonian systems and their connection to passivity N2 - Port-Hamiltonian (pH) systems provide a powerful tool for modeling physical systems. Their energy-based perspective allows for the coupling of various subsystems through energy exchange. Another important class of systems, passive systems, are characterized by their inability to generate energy internally. In this paper, we explore first steps towards understanding the equivalence between passivity and the feasibility of port-Hamiltonian realizations in nonlinear systems. Based on our findings, we present a method to construct port-Hamiltonian representations of a passive system if the dynamics and the Hamiltonian are known. Y1 - 2025 U6 - https://doi.org/10.48550/arXiv.2409.06256 ER - TY - JOUR A1 - Breiten, Tobias A1 - Karsai, Attila T1 - Passive feedback control for nonlinear systems N2 - Dynamical systems can be used to model a broad class of physical processes, and conservation laws give rise to system properties like passivity or port-Hamiltonian structure. An important problem in practical applications is to steer dynamical systems to prescribed target states, and feedback controllers combining a regulator and an observer are a powerful tool to do so. However, controllers designed using classical methods do not necessarily obey energy principles, which makes it difficult to model the controller-plant interaction in a structured manner. In this paper, we show that a particular choice of the observer gain gives rise to passivity properties of the controller that are independent of the plant structure. Furthermore, we state conditions for the controller to have a port-Hamiltonian realization and show that a model order reduction scheme can be deduced using the framework of nonlinear balanced truncation. In addition, we propose a novel passivity preserving discrete gradient scheme for the time discretization of passive systems. To illustrate our results, we numerically realize the controller using the policy iteration and compare it to a controller where the observer gain is given by the extended Kalman filter. Y1 - 2025 U6 - https://doi.org/10.48550/arXiv.2502.04987 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 - Aigner, Kevin-Martin A1 - Denzler, Sebastian A1 - Liers, Frauke A1 - Pokutta, Sebastian A1 - Sharma, Kartikey T1 - Scenario Reduction for Distributionally Robust Optimization N2 - Stochastic and (distributionally) robust optimization problems often become computationally challenging as the number of scenarios increases. Scenario reduction is therefore a key technique for improving tractability. We introduce a general scenario reduction method for distributionally robust optimization (DRO), which includes stochastic and robust optimization as special cases. Our approach constructs the reduced DRO problem by projecting the original ambiguity set onto a reduced set of scenarios. Under mild conditions, we establish bounds on the relative quality of the reduction. The methodology is applicable to random variables following either discrete or continuous probability distributions, with representative scenarios appropriately selected in both cases. Given the relevance of optimization problems with linear and quadratic objectives, we further refine our approach for these settings. Finally, we demonstrate its effectiveness through numerical experiments on mixed-integer benchmark instances from MIPLIB and portfolio optimization problems. Our results show that the oroposed approximation significantly reduces solution time while maintaining high solution quality with only minor errors. KW - distributionally robust optimization KW - scenario reduction KW - scenario clustering KW - approximation bounds KW - mixed-integer programming Y1 - 2025 ER - TY - INPR A1 - Strubberg, Lea A1 - Lutz, Annette A1 - Börner, Pascal A1 - Pfetsch, Marc A1 - Skutella, Martin A1 - Klimm, Max T1 - Valid Cuts for the Design of Potential-based Flow Networks N2 - The construction of a cost minimal network for flows obeying physical laws is an important problem for the design of electricity, water, hydrogen, and natural gas infrastructures. We formulate this problem as a mixed-integer non-linear program. Its non-convexity, due to the poten- tial flow, together with the binary variables, indicating the decision to build a connection, make these problems challenging to solve. We develop a novel class of valid inequalities on the fractional relaxations of the bi- nary variables. Further, we show that this class of inequalities can be sep- arated in polynomial for solutions to a fractional relaxation. This makes it possible to incorporate these inequalities into a branch-and-bound al- gorithm. The advantage of these inequalities is lastly demonstrated in a computational study on the design of real-world gas transport networks. KW - potential based flows KW - topology optimization KW - MINLP Y1 - 2025 ER - TY - INPR A1 - Bock de Barillas, Paulina A1 - Hante, Falk A1 - Hintermüller, Michael T1 - Exact relaxation in optimal switching control for the heat equation N2 - We consider an optimal control problem for the heat equation as a prototypical parabolic partial differential equation with a non-convex control mechanism of the form continuous-or-off. We model this fundamental switching mechanism as the product of a classically continuous and a binary control both in the control term of the dynamics and in the objective. A total variation regularization is added to the cost in order to restrict the number of switching times. This renders the problem as a mixed-integer non-linear PDE-constrained problem. We discuss well-posedness of the problem and present an exact relaxation result for a linearized and a trust-region type penalized problem. The exactness result is constructive and provides a way to numerically compute mixed-integer optimal solutions from the optimality conditions of an associated PDE-constrained problem without integer restrictions. It lays a foundation for a new class of sequential relaxation algorithms to solve the considered class of mixed-integer control problems. This is demonstrated numerically by showcasing a descent step in the presence of binary restrictions. Y1 - 2025 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 - INPR A1 - Breiten, Tobias A1 - Burela, Shubhaditya A1 - Schulze, Philipp T1 - Optimal Control for a Class of Linear Transport-Dominated Systems via the Shifted Proper Orthogonal Decomposition N2 - Solving optimal control problems for transport-dominated partial differential equations (PDEs) can become computationally expensive, especially when dealing with high-dimensional systems. To overcome this challenge, we focus on developing and deriving reduced-order models that can replace the full PDE system in solving the optimal control problem. Specifically, we explore the use of the shifted proper orthogonal decomposition (POD) as a reduced-order model, which is particularly effective for capturing high-fidelity, low-dimensional representations of transport-dominated phenomena. Furthermore, we propose two distinct frameworks for addressing these problems: one where the reduced-order model is constructed first, followed by optimization of the reduced system, and another where the original PDE system is optimized first, with the reduced-order model subsequently applied to the optimality system. We consider a 1D linear advection equation problem and compare the computational performance of the shifted POD method against conventional methods like the standard POD when the reduced-order models are used as surrogates within a backtracking line search. Y1 - 2025 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 - Filipkovska, Maria T1 - Qualitative analysis of nonregular differential-algebraic equations and the dynamics of gas networks N2 - Conditions for the existence, uniqueness and boundedness of global solutions, as well as ultimate boundedness of solutions, and conditions for the blow-up of solutions of nonregular semilinear differential-algebraic equations have been obtained. An example demonstrating the application of the obtained results has been considered. Isothermal models of gas networks have been proposed as applications. KW - nonregular differential-algebraic equation KW - degenerate differential equation KW - singular pencil KW - gas network KW - global solvability Y1 - 2023 U6 - https://doi.org/https://doi.org/10.15407/mag19.04.719 VL - 19 IS - 4 SP - 719 EP - 765 ER - TY - CHAP A1 - Börner, Pascal A1 - Pfetsch, Marc E. A1 - Ulbrich, Stefan T1 - Modeling and optimization of gas mixtures on networks N2 - This paper presents a model for the mixture of gases on networks in the stationary case. The model is based on an equation of state for the mixture, the stationary isothermal Euler equations and coupling conditions for the flow and mixture. The equation of state or pressure law is based on the change of the speed of sound in a mixture of gases. We use this model to solve stationary gas flow problems to global optimality on large networks and present computational results. Y1 - 2024 ER - TY - INPR A1 - Bernhard, Daniela A1 - Liers, Frauke A1 - Stingl, Michael T1 - Robust chance-constrained optimization with discrete distributions N2 - Typically, probability distributions that generate uncertain parameters cannot be measured exactly in practice. As a remedy, distributional robustness determines optimized decisions that are protected in a robust fashion against all probability distributions in some appropriately chosen ambiguity set. In this work, we consider robust joint chance-constrained optimization problems and focus on discrete probability distributions. Many methods for this kind of problems study convex or even linear constraint functions. In contrast, we introduce a practically efficient scenario-based bundle method without convexity assumptions on the constraint functions. We start by deriving an approximation problem to the original robust chance-constrained version by using smoothing and penalization techniques that build on our former work on chance-constrained optimization. Our convergence results with respect to the smoothing approximation and well-known results for penalty approximations suggest replacing the original problem with the approximation problem for large smoothing and penalty parameters. Our scenario-based bundle method starts by solving the approximation problem with a bundle method, and then uses the bundle solution to decide which scenarios to include in a scenario-expanded formulation. This formulation is a standard nonlinear optimization problem. Our approach is guaranteed to find feasible solutions. Furthermore, in the numerical experiments on real-world gas transport problems with uncertain demands, we mostly find globally optimal solutions. Comparing these results to the classical robust reformulations for ambiguity sets consisting of confidence intervals and Wasserstein balls, we observe that the scenario-based bundle method typically outperforms solving the classical reformulation directly. Y1 - 2024 ER - TY - INPR A1 - Giesselmann, Jan A1 - Kunkel, Teresa T1 - Identification of minimal number of measurements allowing synchronization of a nodal observer for the wave equation N2 - We study a state estimation problem for a 2x2 linear hyperbolic system on networks with eigenvalues with opposite signs. The system can be seen as a simplified model for gas flow through gas networks. For this system we construct an observer system based on nodal measurements and investigate the convergence of the state of the observer system towards the original system state. We assume that measurements are available at the boundary nodes of the network and identify the minimal number of additional measurements in the network that are needed to guarantee synchronization of the observer state towards the original system state. It turns out that for tree-shaped networks boundary measurements suffice to guarantee exponential synchronization, while for networks that contain cycles synchronization can be guaranteed if and only if at least one measurement point is added in each cycle. This is shown for a system without source term and for a system with linear friction term. Y1 - 2024 ER - TY - INPR A1 - Wiertz, Ann-Kathrin A1 - Walther, Andrea A1 - Zöttl, Gregor T1 - Strategic Retailers in the Energy Sector N2 - We propose a general framework which allows to analyze the strategic interaction of retail companies, where customers choose retail contracts over a longer period of time based on price and non-price characteristics of retail contracts. We allow for many, possibly asymmetric retailers which can offer fixed price tariffs or dynamic prices, as typically observed in energy markets. Our framework considers uncertainties and allows for price-responsive consumption choices of customers. We analytically characterize all resulting market equilibria for the general asymmetric setting. Based on those results we then propose a solution algorithm which is capable to determine all resulting equilibria. For the case of symmetric retailers we provide analytical comparisons of the different tariff structures. To show the applicability of our framework and our algorithm to real-world instances, we calibrate it to data of the German retail electricity market. Our results show, that firms profits remain unchanged but consumer surplus and welfare increase when switching from fixed price tariffs to real-time pricing. This effect is more pronounced under higher wholesale price fluctuations. Finally we also propose a surrogate, reduced order model, which is shown to be equivalently capable to quantify the welfare difference of the different tariffs. Y1 - ER - TY - INPR A1 - Giannakopoulos, Yiannis A1 - Hahn, Johannes T1 - Discrete Single-Parameter Optimal Auction Design N2 - We study the classic single-item auction setting of Myerson, but under the assumption that the buyers' values for the item are distributed over "finite" supports. Using strong LP duality and polyhedral theory, we rederive various key results regarding the revenue-maximizing auction, including the characterization through virtual welfare maximization and the optimality of deterministic mechanisms, as well as a novel, generic equivalence between dominant-strategy and Bayesian incentive compatibility. Inspired by this, we abstract our approach to handle more general auction settings, where the feasibility space can be given by arbitrary convex constraints, and the objective is a linear combination of revenue and social welfare. We characterize the optimal auctions of such systems as generalized virtual welfare maximizers, by making use of their KKT conditions, and we present an analogue of Myerson's payment formula for general discrete single-parameter auction settings. Additionally, we prove that total unimodularity of the feasibility space is a sufficient condition to guarantee the optimality of auctions with integral allocation rules. Finally, we demonstrate this KKT approach by applying it to a setting where bidders are interested in buying feasible flows on trees with capacity constraints, and provide a combinatorial description of the (randomized, in general) optimal auction. 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 - INPR A1 - Lange, Christian T1 - Modeling and Optimal Control of the Flow of a Gas Mixture N2 - We consider the Euler equations for a pipeline flow of a mixture of two gases. An important application is hydrogen blending. Existence and uniqueness of semi-global solutions is shown and possible boundary conditions are analyzed. Secondly, we consider classes of associated optimal control problems and show existence of solutions. KW - Euler Equations KW - quasilinear hyperbolic system KW - gas transport modeling KW - hydrogen blending Y1 - 2024 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 - INPR A1 - Schuster, Michael T1 - On the Convergence of Optimization Problems with Kernel Density Estimated Probabilistic Constraints N2 - Uncertainty plays a significant role in applied mathematics and probabilistic constraints are widely used to model uncertainty in various fields, even if probabilistic constraints often demand computational challenges. Kernel density estimation (KDE) provides a data-driven approach for properly estimating probability density functions and efficiently evaluate corresponding probabilities. In this paper, we investigate optimization problems with probabilistic constraints, where the probabilities are approximated using a KDE approach. We establish sufficient conditions under which the solution of the KDE approximated optimization problem converges to the solution of the original problem as the sample size goes to infinity. The main results of this paper include three theorems: (1) For sufficiently large sample sizes, the solution of the original problem is also a solution of the approximated problem, if the probabilistic constraint is passive; (2) The limit of a convergent sequence of solutions of the approximated problems is a solution of the original problem, if the KDE uniformly converges; (3) We provide sufficient conditions for the existence of a convergent sequence of solutions of the approximated problems. KW - Probabilistic Constrained Optimization KW - Stochastic Optimization KW - Chance Constraints KW - Kernel Density Estimation KW - Convergence Analysis Y1 - 2024 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 - TY - INPR A1 - Kuchlbauer, Martina T1 - Outer approximation for generalized convex mixed-integer nonlinear robust optimization problems N2 - We consider mixed-integer nonlinear robust optimization problems with nonconvexities. In detail, the functions can be nonsmooth and generalized convex, i.e., f°-quasiconvex or f°-pseudoconvex. We propose a robust optimization method that requires no certain structure of the adversarial problem, but only approximate worst-case evaluations. The method integrates a bundle method, for continuous subproblems, into an outer approximation approach. We prove that our algorithm converges and finds an approximately robust optimal solution and propose robust gas transport as a suitable application. Y1 - ER - TY - INPR A1 - Thürauf, Johannes A1 - Grübel, Julia A1 - Schmidt, Martin T1 - Adjustable Robust Nonlinear Network Design under Demand Uncertainties N2 - We study network design problems for nonlinear and nonconvex flow models under demand uncertainties. To this end, we apply the concept of adjustable robust optimization to compute a network design that admits a feasible transport for all, possibly infinitely many, demand scenarios within a given uncertainty set. For solving the corresponding adjustable robust mixed-integer nonlinear optimization problem, we show that a given network design is robust feasible, i.e., it admits a feasible transport for all demand uncertainties, if and only if a finite number of worst-case demand scenarios can be routed through the network. We compute these worst-case scenarios by solving polynomially many nonlinear optimization problems. Embedding this result for robust feasibility in an adversarial approach leads to an exact algorithm that computes an optimal robust network design in a finite number of iterations. Since all of the results are valid for general potential-based flows, the approach can be applied to different utility networks such as gas, hydrogen, or water networks. We finally demonstrate the applicability of the method by computing robust gas networks that are protected from future demand fluctuations. KW - Robust Optimization KW - Nonlinear Flows KW - Potential-based Networks KW - Demand Uncertainties KW - Mixed-integer Nonlinear Optimization Y1 - 2024 ER - TY - INPR A1 - Wilka, Hendrik A1 - Lang, Jens T1 - Adaptive hp-Polynomial Based Sparse Grid Collocation Algorithms for Piecewise Smooth Functions with Kinks N2 - High-dimensional interpolation problems appear in various applications of uncertainty quantification, stochastic optimization and machine learning. Such problems are computationally expensive and request the use of adaptive grid generation strategies like anisotropic sparse grids to mitigate the curse of dimensionality. However, it is well known that the standard dimension-adaptive sparse grid method converges very slowly or even fails in the case of non-smooth functions. For piecewise smooth functions with kinks, we construct two novel hp-adaptive sparse grid collocation algorithms that combine low-order basis functions with local support in parts of the domain with less regularity and variable-order basis functions elsewhere. Spatial refinement is realized by means of a hierarchical multivariate knot tree which allows the construction of localised hierarchical basis functions with varying order. Hierarchical surplus is used as an error indicator to automatically detect the non-smooth region and adaptively refine the collocation points there. The local polynomial degrees are optionally selected by a greedy approach or a kink detection procedure. Three numerical benchmark examples with different dimensions are discussed and comparison with locally linear and highest degree basis functions are given to show the efficiency and accuracy of the proposed methods. Y1 - ER -