Minimax solutions of path-dependent Hamilton--Jacobi equations under weakened assumptions with application to differential games
Optimization and Control
2026-03-18 v1 Analysis of PDEs
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.
Keywords
Cite
@article{arxiv.2603.16168,
title = {Minimax solutions of path-dependent Hamilton--Jacobi equations under weakened assumptions with application to differential games},
author = {Mikhail Gomoyunov},
journal= {arXiv preprint arXiv:2603.16168},
year = {2026}
}