Representation of Hamilton-Jacobi equation in optimal control theory with compact control set

TL;DR

This paper develops a new method to represent Hamilton-Jacobi equations in optimal control with compact control sets, broadening the class of Hamiltonians for which such representations exist, and addresses an open problem from 2005.

Contribution

It introduces a novel approach to construct representations for a wider class of Hamiltonians in optimal control theory, solving an open problem by Rampazzo (2005).

Findings

Established conditions necessary for Hamilton-Jacobi representation existence.

Extended the class of Hamiltonians with known representations.

Reduced certain variational problems to optimal control problems.

Abstract

In this paper we study the existence of sufficiently regular representations of Hamilton-Jacobi equations in optimal control theory with the compact control set. We introduce a new method to construct representations for a wide class of Hamiltonians, wider than it was achieved before. Our result is proved by means of these conditions on Hamiltonian that are necessary for the existence of a representation. In particular, we solve an open problem of Rampazzo (2005). We apply the obtained results to reduce a variational problem to an optimal control problem.