Feedback control of parametrized PDEs via model order reduction and dynamic programming principle

TL;DR

This paper develops a method combining model order reduction, parameter partitioning, and statistical grid construction to efficiently solve high-dimensional optimal control problems for parametrized PDEs using dynamic programming.

Contribution

It introduces a novel approach integrating basis generation, parameter partitioning, and nonuniform grid construction for reduced-order dynamic programming solutions.

Findings

Effective reduction of computational complexity for PDE control problems.

Successful numerical demonstrations in two-dimensional PDEs.

Enhanced accuracy with nonuniform grid adaptation.

Abstract

In this paper we investigate infinite horizon optimal control problems for parametrized partial differential equations. We are interested in feedback control via dynamic programming equations which is well-known to suffer from the curse of dimensionality. Thus, we apply parametric model order reduction techniques to construct low-dimensional subspaces with suitable information on the control problem, where the dynamic programming equations can be approximated. To guarantee a low number of basis functions, we combine recent basis generation methods and parameter partitioning techniques. Furthermore, we present a novel technique to construct nonuniform grids in the reduced domain, which is based on statistical information. Finally, we discuss numerical examples to illustrate the effectiveness of the proposed methods for PDEs in two space dimensions.