Error analysis for POD Approximations of infinite horizon problems via the Dynamic Programming approach

TL;DR

This paper investigates error analysis for POD-based reduced-order models in solving infinite horizon optimal control problems via dynamic programming, addressing high-dimensional challenges and proposing improved sampling strategies.

Contribution

It introduces an a-priori error estimation for POD approximations of HJB equations and suggests a new sampling strategy to enhance reduced-order model accuracy.

Findings

Error bounds for POD approximations of HJB equations

A new sampling strategy improves POD model accuracy

Numerical experiments validate theoretical error estimates

Abstract

In this paper infinite horizon optimal control problems for nonlinear high-dimensional dynamical systems are studied. Nonlinear feedback laws can be computed via the value function characterized as the unique viscosity solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation which stems from the dynamic programming approach. However, the bottleneck is mainly due to the curse of dimensionality and HJB equations are only solvable in a relatively small dimension. Therefore, a reduced-order model is derived for the dynamical system and for this purpose the method of proper orthogonal decomposition (POD) is used. The resulting errors in the HJB equations are estimated by an a-priori error analysis, which suggests a new sampling strategy for the POD method. Numerical experiments illustrates the theoretical findings.