Optimal polynomial feedback laws for finite horizon control problems

TL;DR

This paper introduces a polynomial-based learning method for finite horizon optimal control that mitigates the curse of dimensionality and is validated through practical examples where traditional HJB approaches are infeasible.

Contribution

It presents a novel polynomial approximation technique for finite horizon control problems that avoids the need for global Lipschitz conditions and demonstrates its effectiveness through examples.

Findings

Method converges under less restrictive conditions.

Effective in high-dimensional problems where HJB is infeasible.

Practical examples show improved computational efficiency.

Abstract

A learning technique for finite horizon optimal control problems and its approximation based on polynomials is analyzed. It allows to circumvent, in part, the curse dimensionality which is involved when the feedback law is constructed by using the Hamilton-Jacobi-Bellman (HJB) equation. The convergence of the method is analyzed, while paying special attention to avoid the use of a global Lipschitz condition on the nonlinearity which describes the control system. The practicality and efficiency of the method is illustrated by several examples. For two of them a direct approach based on the HJB equation would be unfeasible.