Two coupled pursuit-evasion games in target-attacker-defender problem

This paper addresses a differential game with three players: a target, an attacker and a defender, where the Attacker aims to capture the Target while avoiding being captured by the Defender, and the Defender tries to defend the Target from being captured by the Attacker while trying to capture the Attacker. There are two coupled pursuit-evasion problems in this game: Attacker-Target and Defender-Attacker. Firstly, a game of kind in this differential game is considered and a so-called barrier is constructed to divide the space into three disjoint regions: the win region of the Attacker, the win region of the team of the Target and Defender and the uncertain region in which neither of three players can win the game. Secondly, a game of degree is considered by introducing a payoff function. Some explicit expressions of optimal strategies for the players are obtained. Finally, the numerical solutions of optimal strategies and the corresponding optimal trajectories for three players with different initial conditions are provided.