A hybrid differential game with switching thermostatic-type dynamics and cost

TL;DR

This paper studies an infinite horizon zero-sum differential game with switching thermostatic dynamics, characterizing value functions via coupled Hamilton-Jacobi-Isaacs equations and providing conditions for equilibrium existence.

Contribution

It introduces a novel approach to analyze differential games with thermostatic switching dynamics using viscosity solutions and coupled PDE systems.

Findings

Proved continuity of value functions in the switching game

Characterized value functions as solutions to coupled Hamilton-Jacobi-Isaacs equations

Provided conditions for the existence of equilibrium in the game

Abstract

In this paper we consider an infinite horizon zero-sum differential game where the dynamics of each player and the running cost are also depending on the evolution of some discrete (switching) variables. In particular, such switching variables evolve according to the switching law of a so-called thermostatic delayed relay, applied to the players' states. We first address the problem of the continuity of both lower and upper value function. Then, by a suitable representation of the problem as a coupling of several exit-time differential games, we characterize those value functions as, respectively, the unique solution of a coupling of several Dirichlet problems for Hamilton-Jacobi-Isaacs equations. The concept of viscosity solutions and a suitable definition of boundary conditions in the viscosity sense is used in the paper. Finally, we give some sufficient conditions for the existence of an equilibrium.