An engage or retreat differential game with an escort region

This paper is motivated by the desire to develop optimal defensive strategies that discourage an attacker from engaging a high-value target. We analyze an Engage or Retreat Differential Game in which one player represents an attacker and the other player represents a defender. The attacker is capable of terminating the game in engagement or retreat. The defender attempts to manipulate the attacker´s choice through the strategic maximization or minimization of the attacker´s cost function. Both players are free to switch strategies at any point during the game. We show that for certain conditions, the defender should cooperate with the attacker so that retreat becomes the most attractive option. A value function constraint prevents the attacker´s retreat trajectory from passing into a region in which engagement would become optimal. This leads to certain regions of constrained retreat which may be thought of as escort regions. In these escort regions, the defender cooperates with the attacker so long as the attacker maintains appropriate separation from the regions of engagement. The theory is illustrated with a numerical example.