Model order reduction approaches for infinite horizon optimal control problems via the HJB equation

TL;DR

This paper explores model order reduction techniques combined with HJB equations to address the curse of dimensionality in infinite horizon optimal control problems for PDEs, comparing methods like POD, Balanced Truncation, and Riccati-based approaches.

Contribution

It introduces and compares multiple model reduction methods tailored for infinite horizon control problems governed by PDEs, highlighting their advantages and limitations.

Findings

POD effectively reduces system dimension with computational efficiency.

Balanced Truncation preserves stability and accuracy.

Riccati-based approach offers a novel reduction technique.

Abstract

We investigate feedback control for infinite horizon optimal control problems for partial differential equations. The method is based on the coupling between Hamilton-Jacobi-Bellman (HJB) equations and model reduction techniques. It is well-known that HJB equations suffer the so called curse of dimensionality and, therefore, a reduction of the dimension of the system is mandatory. In this report we focus on the infinite horizon optimal control problem with quadratic cost functionals. We compare several model reduction methods such as Proper Orthogonal Decomposition, Balanced Truncation and a new algebraic Riccati equation based approach. Finally, we present numerical examples and discuss several features of the different methods analyzing advantages and disadvantages of the reduction methods.