Path-Dependent Hamilton–Jacobi Equations with u-Dependence and Time-Measurable Hamiltonians

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 (Non-local Hamilton–Jacobi–Bellman equations for the stochastic optimal control of path-dependent piecewise deterministic processes, 2024, http://arxiv.org/abs/2408. 02147), where non-local path-dependent Hamilton–Jacobi–Bellman equations associated to the stochastic optimal control of non-Markovian piecewise deterministic processes are studied.