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

TL;DR

This paper develops a new method to represent Hamilton-Jacobi equations in optimal control with unbounded controls, broadening the class of Hamiltonians that can be analyzed without bounded conjugates, and explores implications for value function regularity.

Contribution

Introduces a novel approach for representing Hamilton-Jacobi equations with unbounded control sets, extending previous methods that required bounded conjugates.

Findings

Constructs representations for a wider class of Hamiltonians.

Shows regularity properties of value functions.

Establishes links between variational and optimal control problems.

Abstract

In this paper we study the existence of sufficiently regular representations of Hamilton-Jacobi equations in the optimal control theory with unbounded control set. We use a new method to construct representations for a wide class of Hamiltonians. This class is wider than any constructed before, because we do not require Legendre-Fenchel conjugates of Hamiltonians to be bounded. However, in this case we obtain representations with unbounded control set. We apply the obtained results to study regularities of value functions and correlations between variational and optimal control problems.