Use of approximations of Hamilton-Jacobi-Bellman inequality for solving periodic optimization problems

TL;DR

This paper demonstrates how approximating the Hamilton-Jacobi-Bellman inequality can effectively solve periodic optimization problems by transforming them into finite-dimensional max-min problems, enabling near-optimal control construction.

Contribution

It introduces a novel approximation method for the HJB inequality using finite-dimensional max-min problems, facilitating practical solutions for periodic optimization.

Findings

Approximate solutions can be used to construct near-optimal controls.

The method is validated through a numerical example.

Necessary and sufficient conditions are established via the HJB inequality.

Abstract

We show that necessary and sufficient conditions of optimality in periodic optimization problems can be stated in terms of a solution of the corresponding HJB inequality, the latter being equivalent to a max-min type variational problem considered on the space of continuously differentiable functions. We approximate the latter with a maximin problem on a finite dimensional subspace of the space of continuously differentiable functions and show that a solution of this problem (existing under natural controllability conditions) can be used for construction of near optimal controls. We illustrate the construction with a numerical example.