Perimeter defense game with nonzero capture radius in a circular target

In this paper, the problem of guarding a circular target wherein the Defender is constrained to move along its perimeter and has nonzero capture radius is posed and solved using a differential game theoretic approach. The Perimeter Defense Game is a special case of Pursuit-Evasion Game, where the goal of the pursuer is capturing the evader. In the Perimeter Defense Game the Attacker seeks to reach the perimeter of the circular target, whereas the Defender seek to align itself with the Attacker, thereby ending the game. The Defender has nonzero capture radius, which means that the Defender wins, when the distance between the Attacker and the Defender is smaller than the value of the capture radius. The Perimeter Defense Game can be divided into two cases: Win of Defender and the Win of Attacker scenarios. In the case when the Defender wins, the agents play a zero sum differential game, where the cost/payoff is the Attacker’s terminal distance to the target. In the case when the Attacker wins, the agents play a zero-sum differential game, where the payoff/cost is the distance between the Defender and the Attacker. The analytic solutions of optimal strategies and the winning regions are also presented.