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 - Kannan, Aswin A1 - Kreimeier, Timo A1 - Walther, Andrea T1 - On Solving Nonsmooth Retail Portfolio Maximization Problems Using Active Signature Methods Y1 - 2023 ER - TY - JOUR A1 - Shyshkanova, Ganna A1 - Walther, Andrea T1 - Optimization of a punch shape with a doubly connected contact domain N2 - The objective is to optimize the pressure distribution under a rigid punch having a doubly connected contact domain close to a circular ring and interacting with an elastic half-space. The required design variable is the punch shape. The functional to be minimized is the root-mean-square deviation of the pressure distribution from some given distribution. An analytical technique is developed for solving the problem for the punches with doubly connected shape, by reducing to a sequence of similar problems for the circular ring punches using expansions of the simple layer potential. The method of expansion in terms of a small parameter is used. The simple layer potential expansion is proposed when mapping a doubly connected integration domain onto a circular ring by transforming the integration variables and transforming the coordinates of the pole of the kernel. As a result, a sequence of similar problems was obtained for a circular ring to determine the functions characterizing the distribution of normal pressure under the punch in the form of a non-circular ring, as well as the normal displacements, from where the optimal punch shape is determined. Y1 - 2023 ER - TY - JOUR A1 - Shyshkanova, Ganna A1 - Zaytseva, Tetyana A1 - Zhushman, V A1 - Levchenko, Ntaliia A1 - Korotunova, Olena T1 - Solving three-dimensional contact problems for foundation design in green building N2 - Design of foundations on an elastic base is carried out using the solution of three-dimensional problems of contact interaction. Improving the accuracy of engineering calculations is necessary to ensure economic efficiency and increase energy savings in green building. The problems of indentation of punches with a flat base bounded by doubly connected close to polygonal contact areas are researched in the present work. Small parameter method is used to obtain explicit analytical expressions for the contact pressure distribution and the punch displacement dependence in a simplified form, which is convenient for engineering practice. The found load-displacement dependence satisfies the known inequalities that are valid for an arbitrary contact domain. Also a numerical-analytical method is in consideration. It uses the simple layer potential expansion and successive approximations for the problems accounting roughness of the elastic half-space. Roughness coefficient is considered as a parameter of regularization of the integral equation for the smooth contact problem. The results of both methods coincide with sufficient accuracy. Y1 - 2023 U6 - https://doi.org/10.1088/1742-6596/2609/1/012001 ER - 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 M. A1 - Schmidt, Martin T1 - Gas Transport Network Optimization: Mixed-Integer Nonlinear Models N2 - Although modern societies strive towards energy systems that are entirely based on renewable energy carriers, natural gas is still one of the most important energy sources. This became even more obvious in Europe with Russia's 2022 war against the Ukraine and the resulting stop of gas supplies from Russia. Besides that it is very important to use this scarce resource efficiently. To this end, it is also of significant relevance that its transport is organized in the most efficient, i.e., cost- or energy-efficient, way. The corresponding mathematical optimization models have gained a lot of attention in the last decades in different optimization communities. These models are highly nonlinear mixed-integer problems that are constrained by algebraic constraints and partial differential equations (PDEs), which usually leads to models that are not tractable. Hence, simplifications have to be made and in this chapter, we present a commonly accepted finite-dimensional stationary model, i.e., a model in which the steady-state solutions of the PDEs are approximated with algebraic constraints. For more details about the involved PDEs and the treatment of transient descriptions we refer to Hante and Schmidt (2023). The presented finite-dimensional as well as mixed-integer nonlinear and nonconvex model is still highly challenging if it needs to be solved for real-world gas transport networks. Hence, we also review some classic solution approaches from the literature. KW - Gas networks KW - Mixed-integer nonlinear optimization KW - Mixed-integer linear optimization KW - Nonlinear optimization Y1 - 2023 ER - TY - INPR A1 - Bongarti, Marcelo A1 - Hintermüller, T1 - Optimal boundary control of the isothermal semilinear Euler equation for gas dynamics on a network N2 - The analysis and boundary optimal control of the nonlinear transport of gas on a network of pipelines is considered. The evolution of the gas distribution on a given pipe is modeled by an isothermal semilinear compressible Euler system in one space dimension. On the network, solutions satisfying (at nodes) the Kirchhoff flux continuity conditions are shown to exist in a neighborhood of an equilibrium state. The associated nonlinear optimization problem then aims at steering such dynamics to a given target distribution by means of suitable (network) boundary controls while keeping the distribution within given (state) constraints. The existence of local optimal controls is established and a corresponding Karush-Kuhn-Tucker (KKT) stationarity system with an almost surely non-singular Lagrange multiplier is derived. KW - optimal boundary control KW - gas dynamics KW - gas networks KW - isothermal Euler equation KW - compressible fluid dynamics Y1 - 2023 ER - TY - INPR A1 - Hante, Falk A1 - Kuchler, Christian T1 - Indirect methods for optimal control of parabolic hybrid PDE-dynamical / switching systems using relaxation N2 - We propose a novel algorithmic approach to computationally solve optimal control problems governed by linear parabolic partial differential equations (PDEs) including a state-dependent control-regime switching mechanism. We state an equivalent mixed-integer formulation featuring vanishing constraints (VCs) arising from methods of disjunctive programming. We embed the problem into the class of equilibrium constraints (ECs) by introduction of an additional slack variable. Based on theoretical results associated with Sum-Up-Rounding (SUR) strategies, we proceed with the solution of the related relaxed formulation by an indirect approach. In order to obtain a computationally tractable optimality system, we apply a Moreau-Yosida type penalty approach for the VCs. After a theoretical discussion, we introduce and exert the algorithmic framework founded on a semismooth Newton method. Finally, we communicate computational experiments based on the proposed approach. Y1 - ER -