Lipschitz Continuity Results for Minimax Solutions of Path-Dependent Hamilton--Jacobi Equations

TL;DR

This paper establishes Lipschitz continuity properties for minimax solutions of path-dependent Hamilton--Jacobi equations, which are relevant in optimal control and differential games involving delay systems.

Contribution

It proves Lipschitz regularity of minimax solutions for a class of path-dependent Hamilton--Jacobi equations under specific assumptions.

Findings

Solutions are Lipschitz continuous in time and functional variables.

Lipschitz conditions are formulated with respect to uniform and special norms.

Results apply to value functionals in control problems and differential games.

Abstract

We consider a Cauchy problem for a (first-order) path-dependent Hamilton--Jacobi equation with coinvariant derivatives and a right-end boundary condition. Such problems arise naturally in the study of properties of the value functional in (deterministic) optimal control problems and differential games for time-delay systems. We prove that, under certain assumptions on the Hamiltonian and the boundary functional, a minimax (generalized) solution of the Cauchy problem satisfies Lipschitz conditions both in time and functional variables. In the latter case, Lipschitz conditions are formulated with respect to the uniform norm and some special norm of the space of continuous functions.