Representation of convex Hamilton-Jacobi equations in optimal control theory

TL;DR

This paper investigates how convex Hamilton-Jacobi equations can be represented through optimal control problems, introducing a new method to construct broader classes of such representations and analyzing their stability.

Contribution

It presents a novel approach to represent convex Hamilton-Jacobi equations via optimal control, extending previous results to wider classes of Hamiltonians with stability analysis.

Findings

New method for representing a broader class of Hamiltonians

Construction of representations with both compact and noncompact control sets

Proof of stability of the constructed representations

Abstract

In the paper we study the following problem: given a Hamilton-Jacobi equation where the Hamiltonian is convex with respect to the last variable, are there any optimal control problems representing it? In other words, we search for an appropriately regular dynamics and a Lagrangian that represents the Hamiltonian with given properties. This problem was lately researched by Frankowska-Sedrakyan (2014) and Rampazzo (2005). We introduce a new method to construct a representation of a wide class of Hamiltonians, wider than it was achieved before. Actually, we get two types of representations: with compact and noncompact control set, depending on regularity of the Hamiltonian. We conclude the paper by proving the stability of representations.