TY - INPR A1 - Gugat, Martin A1 - Habermann, Jens A1 - Hintermüller, Michael A1 - Huber, Olivier T1 - Constrained exact boundary controllability of a semilinear model for pipeline gas flow N2 - While the quasilinear isothermal Euler equations are an excellent model for gas pipeline flow, the operation of the pipeline flow with high pressure and small Mach numbers allows us to obtain approximate solutions by a simpler semilinear model. We provide a derivation of the semilinear model that shows that the semilinear model is valid for sufficiently low Mach numbers and sufficiently high pressures. We prove an existence result for continuous solutions of the semilinear model that takes into account lower and upper bounds for the pressure and an upper bound for the magnitude of the Mach number of the gas flow. These state constraints are important both in the operation of gas pipelines and to guarantee that the solution remains in the set where the model is physically valid. We show the constrained exact boundary controllability of the system with the same pressure and Mach number constraints. Y1 - 2021 ER - TY - INPR A1 - Domschke, Pia A1 - Hiller, Benjamin A1 - Lang, Jens A1 - Mehrmann, Volker A1 - Morandin, Riccardo A1 - Tischendorf, Caren T1 - Gas Network Modeling: An Overview N2 - With this overview we want to provide a compilation of different models for the description of gas flow in networks in order to facilitate the introduction to the topic. Special attention is paid to the hierarchical structure inherent to the modeling, and the detailed description of individual components such as valves and compressors. Also included are network model classes based on purely algebraic relations, and energy-based port-Hamiltonian models. A short overview of basic numerical methods and concepts for the treatment of hyperbolic balance equations is also given. We do not claim completeness and refer in many places to the existing literature. The idea of a model catalog came to us in the context of the application for the CRC/Transregio 154 ``Mathematical modeling, simulation and optimization using the example of gas networks''. The present English translation is an extension from [P. Domschke, B. Hiller, J. Lang, and C. Tischendorf. Modellierung von Gasnetzwerken: Eine Übersicht. Preprint, TRR 154, 2017]. At this point we would like to thank the DFG for its support. Y1 - 2021 ER - TY - JOUR A1 - Gugat, Martin A1 - Hante, Falk A1 - Jin, Li T1 - Closed loop control of gas flow in a pipe: Stability for a transient model JF - at - Automatisierungstechnik N2 - This contribution focuses on the analysis and control of friction-dominated flow of gas in pipes. The pressure in the gas flow is governed by a partial differential equation that is a doubly nonlinear parabolic equation of p-Laplace type, where p=2/3. Such equations exhibit positive solutions, finite speed of propagation and satisfy a maximum principle. The pressure is fixed on one end (upstream), and the flow is specified on the other end (downstream). These boundary conditions determine a unique steady equilibrium flow. We present a boundary feedback flow control scheme, that ensures local exponential stability of the equilibrium in an L2-sense. The analysis is done both for the pde system and an ode system that is obtained by a suitable spatial semi-discretization. The proofs are based upon suitably chosen Lyapunov functions. Y1 - 2020 ER - TY - JOUR A1 - Hajian, Soheil A1 - Hintermüller, Michael A1 - Ulbrich, Stefan T1 - Total variation diminishing schemes in optimal control of scalar conservation laws JF - IMA Journal of Numerical Analysis N2 - In this paper, optimal control problems subject to a nonlinear scalar conservation law are studied. Such optimal control problems are challenging both at the continuous and at the discrete level since the control-to-state operator poses difficulties as it is, e.g., not differentiable. Therefore discretization of the underlying optimal control problem should be designed with care. Here the discretize-then-optimize approach is employed where first the full discretization of the objective function as well as the underlying PDE is considered. Then, the derivative of the reduced objective is obtained by using an adjoint calculus. In this paper total variation diminishing Runge-Kutta (TVD-RK) methods for the time discretization of such problems are studied. TVD-RK methods, also called strong stability preserving (SSP), are originally designed to preserve total variation of the discrete solution. It is proven in this paper that providing an SSP state scheme, is enough to ensure stability of the discrete adjoint. However requiring SSP for both discrete state and adjoint is too strong. Also approximation properties that the discrete adjoint inherits from the discretization of the state equation are studied. Moreover order conditions are derived. In addition, optimal choices with respect to CFL constant are discussed and numerical experiments are presented. Y1 - 2017 U6 - https://doi.org/10.20347/WIAS.PREPRINT.2383 VL - 39 SP - 105 EP - 140 ER - TY - JOUR A1 - Papafitsoros, Konstantinos A1 - Hintermüller, Michael A1 - Rautenberg, Carlos T1 - Analytical aspects of spatially adapted total variation regularisation JF - Journal of Mathematical Analysis and Applications N2 - In this paper we study the structure of solutions of the one dimensional weighted total variation regularisation problem, motivated by its application in signal recovery tasks. We study in depth the relationship between the weight function and the creation of new discontinuities in the solution. A partial semigroup property relating the weight function and the solution is shown and analytic solutions for simply data functions are computed. We prove that the weighted total variation minimisation problem is well-posed even in the case of vanishing weight function, despite the lack of coercivity. This is based on the fact that the total variation of the solution is bounded by the total variation of the data, a result that it also shown here. Finally the relationship to the corresponding weighted fidelity problem is explored, showing that the two problems can produce completely different solutions even for very simple data functions. Y1 - 2017 VL - 454 SP - 891 EP - 935 ER - TY - JOUR A1 - Keil, Tobias A1 - Hintermüller, Michael A1 - Wegner, Donat T1 - Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system with non-matched fluid densities JF - SIAM Journal on Control and Optimization N2 - This paper is concerned with the distributed optimal control of a time-discrete Cahn– Hilliard/Navier–Stokes system with variable densities. It focuses on the double-obstacle potential which yields an optimal control problem for a family of coupled systems in each time instant of a variational inequality of fourth order and the Navier–Stokes equation. By proposing a suitable time- discretization, energy estimates are proved and the existence of solutions to the primal system and of optimal controls is established for the original problem as well as for a family of regularized problems. The latter correspond to Moreau–Yosida type approximations of the double-obstacle potential. The consistency of these approximations is shown and first order optimality conditions for the regularized problems are derived. Through a limit process with respect to the regularization parameter, a stationarity system for the original problem is established. The resulting system corresponds to a function space version of C-stationarity which is a special notion of stationarity for MPECs. Y1 - 2017 VL - 55 SP - 1954 EP - 1989 ER - TY - JOUR A1 - Hintermüller, Michael A1 - Rautenberg, Carlos A1 - Mohammadi, Masoumeh A1 - Kanitsar, Martin T1 - Optimal sensor placement: A robust approach JF - SIAM Journal on Control and Optimization N2 - We address the problem of optimally placing sensor networks for convection-diffusion processes where the convective part is perturbed. The problem is formulated as an optimal control problem where the integral Riccati equation is a constraint and the design variables are sensor locations. The objective functional involves a term associated to the trace of the solution to the Riccati equation and a term given by a constrained optimization problem for the directional derivative of the previous quantity over a set of admissible perturbations. The paper addresses the existence of the derivative with respect to the convective part of the solution to the Riccati equation, the well-posedness of the optimization problem and finalizes with a range of numerical tests. Y1 - 2017 VL - 55 SP - 3609 EP - 3639 ER - TY - JOUR A1 - Gonzalez Grandon, Tatiana A1 - Heitsch, Holger A1 - Henrion, Rene T1 - A joint model of probabilistic/robust constraints for gas transport management in stationary networks JF - Computational Management Science N2 - We present a novel mathematical algorithm to assist gas network operators in managing uncertainty, while increasing reliability of transmission and supply. As a result, we solve an optimization problem with a joint probabilistic constraint over an infinite system of random inequalities. Such models arise in the presence of uncertain parameters having partially stochastic and partially non-stochastic character. The application that drives this new approach is a stationary network with uncertain demand (which are stochastic due to the possibility of fitting statistical distributions based on historical measurements) and with uncertain roughness coefficients in the pipes (which are uncertain but non-stochastic due to a lack of attainable measurements). We study the sensitivity of local uncertainties in the roughness coefficients and their impact on a highly reliable network operation. In particular, we are going to answer the question, what is the maximum uncertainty that is allowed (shaping a 'maximal' uncertainty set) around nominal roughness coefficients, such that random demands in a stationary gas network can be satisfied at given high probability level for no matter which realization of true roughness coefficients within the uncertainty set. One ends up with a constraint, which is probabilistic with respect to the load of gas and robust with respect to the roughness coefficients. We demonstrate how such constraints can be dealt with in the framework of the so-called spheric-radial decomposition of multivariate Gaussian distributions. The numerical solution of a corresponding optimization problem is illustrated. The results might assist the network operator with the implementation of cost-intensive roughness measurements. KW - chance constraint KW - robust constraint KW - uncertainty set KW - spheric-radial decomposition Y1 - 2017 U6 - https://doi.org/10.1007/s10287-017-0284-7 VL - 14 SP - 443 EP - 460 ER - TY - INPR A1 - Huck, Christoph A1 - Tischendorf, Caren T1 - Perturbation analysis of hyperbolic PDAEs describing flow networks Y1 - 2017 ER - TY - CHAP A1 - Benner, Peter A1 - Grundel, Sarah A1 - Himpe, Christian A1 - Huck, Christoph A1 - Streubel, Tom A1 - Tischendorf, Caren T1 - Gas Network Benchmark Models T2 - Applications of Differential-Algebraic Equations: Examples and Benchmarks N2 - The simulation of gas transportation networks becomes increasingly more important as its use-cases broaden to more complex applications. Classically, the purpose of the gas network was the transportation of predominantly natural gas from a supplier to the consumer for long-term scheduled volumes. With the rise of renewable energy sources, gas-fired power plants are often chosen to compensate for the fluctuating nature of the renewables, due to their on-demand power generation capability. Such an only short-term plannable supply and demand setting requires sophisticated simulations of the gas network prior to the dispatch to ensure the supply of all customers for a range of possible scenarios and to prevent damages to the gas network. In this work we describe the modeling of gas networks and present benchmark systems to test implementations and compare new or extended models. Y1 - 2017 U6 - https://doi.org/10.1007/11221_2018_5 VL - DAE-FORUM SP - 171 EP - 197 PB - Springer CY - Cham ER -