TY - JOUR A1 - Gugat, Martin A1 - Leugering, Günter A1 - Martin, Alexander A1 - Schmidt, Martin A1 - Sirvent, Mathias A1 - Wintergerst, David T1 - Towards Simulation Based Mixed-Integer Optimization with Differential Equations JF - Networks N2 - We propose a decomposition based method for solving mixed-integer nonlinear optimization problems with “black-box” nonlinearities, where the latter, e.g., may arise due to differential equations or expensive simulation runs. The method alternatingly solves a mixed-integer linear master problem and a separation problem for iteratively refining the mixed-integer linear relaxation of the nonlinear equalities. The latter yield nonconvex feasible sets for the optimization model but we have to restrict ourselves to convex and monotone constraint functions. Under these assumptions, we prove that our algorithm finitely terminates with an approximate feasible global optimal solution of the mixed integer nonlinear problem. Additionally, we show the applicability of our approach for three applications from optimal control with integer variables, from the field of pressurized flows in pipes with elastic walls, and from steady-state gas transport. For the latter we also present promising numerical results of our method applied to real-world instances that particularly show the effectiveness of our method for problems defined on networks. KW - Mixed-Integer Optimization KW - Simulation Based Optimization KW - Optimization with Differential Equations KW - Decomposition Method KW - Gas Transport Networks Y1 - 2018 U6 - https://doi.org/10.1002/net.21812 ER - TY - JOUR A1 - Gugat, Martin A1 - Steffensen, Sonja ED - Trelat, Emmanuel T1 - DYNAMIC BOUNDARY CONTROL GAMES WITH NETWORKS OF STRINGS JF - ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV) N2 - Consider a star-shaped network of strings. Each string is governed by the wave equation. At each boundary node of the network there is a player that performs Dirichlet boundary control action and in this way influences the system state. At the central node, the states are coupled by algebraic conditions in such a way that the energy is conserved. We consider the corresponding antagonistic game where each player minimizes a certain quadratic objective function that is given by the sum of a control cost and a tracking term for the final state. We prove that under suitable assumptions a unique Nash equilibrium exists and give an explicit representation of the equilibrium strategies. Y1 - 2018 U6 - https://doi.org/10.1051/cocv/2017082 ER - TY - JOUR A1 - Gugat, Martin A1 - Rosier, Lionel A1 - Perrollaz, Vincent ED - Arendt, W. ED - Pierre, M. T1 - Boundary stabilization of quasilinear hyperbolic systems of balance laws: exponential decay for small source terms JF - Journal of Evolution Equations N2 - We investigate the long-time behaviour of solutions of quasilinear hyperbolic systems with transparent boundary conditions when small source terms are incorporated in the system. Even if the finite-time stability of the system is not preserved, it is shown here that an exponential convergence towards the steady state still holds with a decay rate which is proportional to the logarithm of the amplitude of the source term. The result is stated for a system with dynamical boundary conditions in order to deal with initial data that are free of any compatibility condition. The proof of the existence and uniqueness of a solution defined for all positive times is also provided in this paper. Y1 - 2018 U6 - https://doi.org/10.1007/s00028-018-0449-z ER -