Deterministic Differential Games in Infinite Horizon Involving Continuous and Impulse Controls

TL;DR

This paper studies a two-player deterministic differential game with both continuous and impulse controls over an infinite horizon, establishing existence, uniqueness, and the value of the game through PDE analysis.

Contribution

It introduces a framework for analyzing differential games with combined control types and proves the existence and uniqueness of solutions to the associated HJBI equations.

Findings

Existence and uniqueness of viscosity solutions to HJBI PDEs.

Coincidence of upper and lower value functions under Isaacs condition.

Framework accommodates general impulse costs depending on the system state.

Abstract

We consider a two-player zero-sum deterministic differential game where each player uses both continuous and impulse controls in infinite-time horizon. We assume that the impulses supposed to be of general term and the costs depend on the state of the system. We use the dynamic programming principle and viscosity solutions approach to show existence and uniqueness of a solution for the Hamilton-Jacobi-Bellman-Isaacs (HJBI) partial differential equations (PDEs) of the game. We prove under Isaacs condition that the upper and lower value functions coincide.