Path-dependent Hamilton-Jacobi equations with u-dependence and time-measurable Hamiltonians

TL;DR

This paper proves existence and uniqueness of minimax solutions for a broad class of path-dependent Hamilton-Jacobi equations with time-measurable Hamiltonians, advancing optimal control theory for delay differential equations.

Contribution

It introduces a general framework for path-dependent Hamilton-Jacobi equations with u-dependence and time-measurable Hamiltonians, including applications to delay differential control problems.

Findings

Established existence and uniqueness of solutions.

Applied results to delay functional differential equations.

Provided foundational results for non-Markovian stochastic control.

Abstract

We establish existence and uniqueness of minimax solutions for a fairly general class of path-dependent Hamilton-Jacobi equations. In particular, the relevant Hamiltonians can contain the solution and they only need to be measurable with respect to time. We apply our results to optimal control problems of (delay) functional differential equations with cost functionals that have discount factors and with time-measurable data. Our main results are also crucial for our companion paper Bandini and Keller [arXiv preprint arXiv:2408.02147 (2024)], where non-local path-dependent Hamilton-Jacobi-Bellman equations associated to the stochastic optimal control of non-Markovian piecewise deterministic processes are studied.