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arXiv · 2404.07015

POD Suboptimal Control of Evolution Problems: Theory and Applications

Stefan Banholzer·Dennis Beermann·Luca Mechelli·Stefan Volkwein
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Abstract

The work is organized as follows. First an introduction is given in Chapter 1. In Chapter 2 we introduce the POD method in finite and infinite-dimensional Hilbert spaces and discuss various applications. Chapter 3 is devoted to to POD-based Galerkin schemes for evolution problems. Mainly, we study linear problems taking different discretization methods into account. We provide a certified a-priori and a-posteriori error analysis. Furthermore, the numerical realizations are explained and illustrated by test examples. Quadratic programming problems governed by liner evolution problems are investigated in Chapter 4. As in Chapter 3 we present a certified a-priori and a-posteriori error analysis. Moreover, we discuss basis update strategies. In Chapter 5 we give an outlook to further directions in reduced-order modeling in optimal control and optimization. More precisely, a nonlinear optimal control problem is studied. Moreover, state-constrained optimization problems are solved by a tailored combination of primal-dual active set methods and POD-aesed reduced-order modeling. Furthermore, POD Galerkin methods for multiobjective optimal control problems are investigated. Finally, some required theoretical foundations are summarized in the appendix.

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BibTeXRIS

Stefan Banholzer, Dennis Beermann, Luca Mechelli, Stefan Volkwein. 2024-04-10. POD Suboptimal Control of Evolution Problems: Theory and Applications. https://arxiv.org/abs/2404.07015

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