A differential game with exit costs

TL;DR

This paper analyzes a differential game involving two players with individual dynamics and exit costs, proving the continuity of game values and their characterization as viscosity solutions of Hamilton-Jacobi-Isaacs equations.

Contribution

It introduces a novel framework for differential games with multiple exit costs and establishes the continuity and uniqueness of the value functions under domain constraints.

Findings

Values are continuous under suitable hypotheses.

Values are the unique viscosity solutions of a Dirichlet problem.

Existence of non-anticipating strategies respecting domain constraints.

Abstract

We study a differential game where two players separately control their own dynamics, pay a running cost, and moreover pay an exit cost (quitting the game) when they leave a fixed domain. In particular, each player has its own domain and the exit cost consists of three different exit costs, depending whether either the first player only leaves its domain, or the second player only leaves its domain, or they both simultaneously leave their own domain. We prove that, under suitable hypotheses, the lower and upper value are continuous and are, respectively, the unique viscosity solution of a suitable Dirichlet problem for a Hamilton-Jacobi-Isaacs equation. The continuity of the values relies on the existence of suitable non-anticipating strategies respecting the domain-constraint. This problem is also treated in this work.