A Zero-Sum Differential Game with Exit Time

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 Dirichlet problem for the Hamilton-Jacobi equation associated with the game with an exit time. We prove the existence of the value function for the examined problem and construct suboptimal feedback strategies under the assumption that the associated Dirichlet problem for the Hamilton-Jacobi equation admits a viscosity/minimax solution. Additionally, we derive a sufficient condition of the existence result to this Dirichlet problem.