Equivalence of minimax and viscosity solutions of path-dependent Hamilton-Jacobi equations

In the paper, we consider a path-dependent Hamilton-Jacobi equation with coinvariant derivatives over the space of continuous functions. Such equations arise from optimal control problems and differential games for time-delay systems. We study generalized solutions of the considered Hamilton-Jacobi equation both in the minimax and in the viscosity sense. A minimax solution is defined as a functional which epigraph and subgraph satisfy certain conditions of weak invariance, while a viscosity solution is defined in terms of a pair of in-equalities for coinvariant sub-and supergradients. We prove that these two notions are equivalent, which is the main re-sult of the paper. As a corollary, we obtain comparison and uniqueness results for viscosity solutions of a Cauchy problem for the considered Hamilton-Jacobi equation and a right-end boundary condition. The proof of the main result is based on a certain property of the coinvariant subdifferential. To establish this property, we develop a technique going back to the proofs of multidirectional mean-value inequalities. In particular, the absence of the local compactness property of the underlying continuous function space is overcome by using Borwein-Preiss variational principle with an appropriate gauge-type functional. (c) 2023 Elsevier Inc. All rights reserved.

Automated summary

The paper discusses a path-dependent Hamilton-Jacobi equation with coinvariant derivatives over the space of continuous functions. It studies generalized solutions of the equation in both the minimax and viscosity senses, and proves the equivalence between these two notions. The paper also obtains comparison and uniqueness results for viscosity solutions of a Cauchy problem with a right-end boundary condition.