A Zero-Sum Differential Game with Exit Time

TL;DR

This paper studies a zero-sum differential game where the payoff depends on the exit time from a domain, including cases with a boundary 'lifeline' that yields infinite payoff, extending classical time-optimal problems.

Contribution

It establishes the existence of the value function and constructs suboptimal strategies using viscosity solutions of the Hamilton-Jacobi equation for the exit time game.

Findings

Proves the existence of the value function for the game.

Constructs suboptimal feedback strategies.

Provides a sufficient condition for the Dirichlet problem's solvability.

Abstract

The paper is concerned with a zero-sum differential game in the case where a payoff is determined by the exit time, that is, the first time when the system leaves the game domain. Additionally, we assume that a part of domain's boundary is a lifeline where the payoff is infinite. Hereby, the examined problem generalizes the well-known time-optimal problem as well as time-optimal problem with lifeline. The main result of the paper relies on the solution to the Direchlet problem for the Hamilton-Jacobi equation associated with the game with exit time. We prove the existence of the value function for examined problem and construct suboptimal feedback strategies under assumption that the associated Dirichlet problem for the Hamilton-Jacobi equation admits a viscosity/minimax solution. Additionally, we derive a sufficient condition of existence result to this Dirichlet problem.