POD Suboptimal Control of Evolution Problems: Theory and Applications

TL;DR

This paper develops and analyzes Proper Orthogonal Decomposition (POD) methods for evolution problems, including linear and nonlinear optimal control, with error analysis, basis update strategies, and applications to constrained optimization.

Contribution

It introduces POD-based Galerkin schemes for evolution problems, provides error analysis, and extends the approach to nonlinear optimal control and constrained problems.

Findings

Certified a-priori and a-posteriori error estimates for linear problems.

Numerical illustrations demonstrating the effectiveness of POD methods.

Extension of POD techniques to nonlinear optimal control and state-constrained problems.

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.