# Minimax and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations for Time-Delay Systems

## TL;DR

This paper investigates Hamilton-Jacobi-Bellman equations for time-delay systems, establishing the existence, uniqueness, and equivalence of minimax and viscosity solutions to the value functional in a delay differential control problem.

## Contribution

It introduces a framework for analyzing Hamilton-Jacobi-Bellman equations with delay, proving the equivalence of minimax and viscosity solutions for such systems.

## Key findings

- Existence of minimax and viscosity solutions
- Uniqueness of these solutions
- Solutions coincide with the value functional

## Abstract

The paper deals with a Bolza optimal control problem for a dynamical system which motion is described by a delay differential equation under an initial condition defined by a piecewise continuous function. For the value functional in this problem, the Cauchy problem for the Hamilton-Jacobi-Bellman equation with coinvariant derivatives is considered. Minimax and viscosity solutions of this problem are studied. It is proved that both of these solutions exist, are unique and coincide with the value functional.