Circular Target Defense Differential Games

Alexander Von Moll, Meir Pachter, Daigo Shishika, Zachariah E. Fuchs

Abstract: In this article, the problem of guarding a circular target wherein the Defender(s) is constrained to move along its perimeter is posed and solved using a differential game theoretic approach. Both the one-Defender and two-Defender scenarios are analyzed and solved. The mobile Attacker seeks to reach the perimeter of the circular target, whereas the Defender(s) seeks to align itself with the Attacker, thereby ending the game. In the former case, the Attacker-wins, and the Attacker and Defender play a zero-sum differential game where the payoff/cost is the terminal angular separation. In the latter case, the Defender(s) wins, and the Attacker and Defender play a zero-sum differential game where the cost/payoff is the Attacker’s terminal distance to the target. This formulation is representative of a scenario in which the Attacker inflicts damage on the target as a function of its terminal distance. The state-feedback equilibrium strategies and Value functions for the Attacker-win and Defender(s)-win scenarios are derived for both the one- and two-Defender cases, thus providing a solution to the Game of Degree. Analytic expressions for the separating surfaces between the various terminal scenarios are derived, thus providing a solution to the Game of Kind. An alternative game is formulated and solved in the case of Attacker-win wherein the Attacker seeks to minimize time to reach the target.