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

TL;DR

This paper proves the equivalence between minimax and viscosity solutions for path-dependent Hamilton-Jacobi equations, providing key comparison and uniqueness results for such equations in control and differential game contexts.

Contribution

It establishes the equivalence of minimax and viscosity solutions for path-dependent Hamilton-Jacobi equations, a novel result in the theory of such equations.

Findings

Proves the equivalence of minimax and viscosity solutions.

Derives comparison and uniqueness results for viscosity solutions.

Develops a new technique using variational principles for non-compact spaces.

Abstract

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 inequalities for coinvariant sub- and super-gradients. We prove that these two notions are equivalent, which is the main result 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 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 guage-type functional.