Engage or Retreat Differential Game with Two Targets

We examine a two player, engage or retreat differential game, in which one player represents a mobile attacker and the opposing player represents a defender protecting two static, high-value targets. The Attacker chooses to terminate the game either by capturing any one of the two high-value targets or retreating to a defined retreat surface. The Defender's sole goal, is to protect the high-value targets. It attempts to persuade the Attacker into retreat by manipulating the integral cost within the Attacker's utility function. We pose this scenario as a non-zero sum differential game and show that the solution of this game can be constructed from the solutions of two related optimization problems. Additional path constraints are imposed on retreat trajectories in order to ensure the Attacker does not move into regions in which it becomes optimal to switch to engagement. The complete solution of the game divides the state-space into two regions with different qualitative agent behaviors. In one region, the Attacker engages the Target and the Defender responds by minimizing the Attacker's utility. In the other region, the Attacker retreats and the Defender responds by maximizing the Attacker's utility.