Optimal strategies for the lifeline differential game with limited lifetime

This paper considers the Lifeline Differential Game with limited lifetime in a plane which is divided by the lifeline into a play region and a goal region. A single evader aims at entering the goal region from the play region by penetrating the lifeline without being captured, while a single pursuer strives to prevent the evader from that by capturing or delaying it with non-zero capture radius. We introduce the concept of the limited lifetime within which the evader has to achieve its goal. Determining the game winner and finding the strategies for two players are challenging due to non-unique terminal conditions and various initial configurations, which is beyond the scope of the classical Hamilton–Jacobi–Isaacs (HJI) approach. An evasion-region based geometric method is proposed to solve this game qualitatively and quantitatively. A new class of Apollonius circles is first introduced and thus a pursuit strategy is proposed for the pursuer to capture the evader at a guaranteed distance of the lifeline. Based on the pursuit strategy, the space of initial states is separated by the so-called barrier into two winning regions which lead to different outcomes of the game. In each winning region, a critical payoff function is designed and the optimal strategy for each player is investigated. The extension to multiplayer cases is also discussed. The presented results are analytical and thus allow for real-time updates.