Viscosity solutions of Hamilton-Jacobi equations for neutral-type systems

TL;DR

This paper develops a theory of viscosity solutions for path-dependent Hamilton-Jacobi equations related to neutral-type systems, establishing existence, uniqueness, and applications to optimal control problems.

Contribution

It introduces new definitions and proves fundamental properties of viscosity solutions for these complex equations, extending the mathematical framework for neutral-type systems.

Findings

Proved existence and uniqueness of viscosity solutions.

Established equivalence of multiple solution definitions.

Applied theory to optimal control problems for neutral systems.

Abstract

The paper deals with path-dependent Hamilton-Jacobi equations with a coinvariant derivative which arise in investigations of optimal control problems and differential games for neutral-type systems in Hale's form. A viscosity (generalized) solution of a Cauchy problem for such equations is considered. The existence, uniqueness, and consistency of the viscosity solution are proved. Equivalent definitions of the viscosity solution, including the definitions of minimax and Dini solutions, are obtained. Application of the results to an optimal control problem for neutral-type systems in Hale's form are given.