Minimax solutions of path-dependent Hamilton--Jacobi equations under weakened assumptions with application to differential games

TL;DR

This paper develops a theory for minimax solutions of path-dependent Hamilton--Jacobi equations with weaker assumptions on the Hamiltonian, establishing existence, uniqueness, and stability, and applies it to zero-sum differential games with time delays.

Contribution

It introduces more general conditions for the Hamiltonian in path-dependent Hamilton--Jacobi equations and proves key properties of minimax solutions, extending previous results.

Findings

Established existence and uniqueness of minimax solutions under weakened assumptions.

Proved stability and consistency of solutions in the path-dependent setting.

Demonstrated the applicability to zero-sum differential games with time delays.

Abstract

We study minimax (generalized) solutions of a Cauchy problem for a (first-order) path-dependent Hamilton--Jacobi equation with co-invariant derivatives under a right-end boundary condition. Under assumptions on the Hamiltonian that are more general than those previously considered in the literature and allow, in particular, a measurable dependence on the first (time) variable, we establish existence, uniqueness, stability, and consistency results for minimax solutions. As an application, we consider a zero-sum differential game for a time-delay system and prove that this game has a value under assumptions more general than the known ones but rather natural being consistent with the Carath\'{e}odory conditions.