TY - INPR A1 - Hante, Falk M. A1 - Schmidt, Martin T1 - Gas Transport Network Optimization: PDE-Constrained Models N2 - The optimal control of gas transport networks was and still is a very important topic for modern economies and societies. Accordingly, a lot of research has been carried out on this topic during the last years and decades. Besides mixed-integer aspects in gas transport network optimization, one of the main challenges is that a physically and technically detailed modeling of transient gas dynamics leads to theoretically and computationally highly demanding models involving nonlinear partial differential equations (PDEs). For further background on the application, historical notes and a detailed discussion of mixed-integer aspects for stationary descriptions we refer to Hante and Schmidt (2023). In this chapter, we focus on the most common modeling approaches concerning transient descriptions, point out the challenges, and summarize important contributions concerning the optimization of the most relevant control parameters for this particular class of problems. KW - Gas networks KW - Partial differential equations KW - Optimal control KW - PDE-constrained optimization KW - Modeling Y1 - 2023 ER - TY - INPR A1 - Hante, Falk A1 - Kuchler, Christian T1 - An Algorithmic Framework for Optimal Control of Hybrid Dynamical System with Parabolic PDEs N2 - We present an algorithmic approach for the computational solution of optimal control problems with hybrid nature governed by linear parabolic PDEs featuring implicit switches. We propose a stepwise reformulation of the original formulation into a more tractable setting via application of methods from disjunctive programming and a time transformation method. After removal of the implicit switching rule at the cost of the introduction of explicit switching variables and vanishing constraints, the connection of the resulting formulation to problems with equilibrium constraints is established and studied. The previous steps in combination with smoothening and a Moreau-Yosida type penalty approach allow the derivation of necessary first order optimality conditions to characterize candidates for optimality to the original system. Following the discussion of each individual reformulation step, we introduce the algorithmic framework founded on a semismooth Newton method. Finally, we report on computational of the proposed framework. Y1 - 2023 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 -