Optimal Solution of a Target Defense Game with Two defenders and a Faster Intrude

Abstract

A target defense game with two defenders and a faster intruder is solved based on the classic differential game theory. In the game, the intruder seeks to enter a circular target area, while the defenders endeavor to capture it outside of the target. Under the faster intruder assumption, the game has two phases, where the optimal trajectories are straight and curved, respectively. In the second phase, a peculiar phenomenon exists where the intruder moves at the edge of one defender’s capture region, yet this defender cannot force capture. Because of this, the terminal states of the game are singular, therefore the standard method of integrating optimal trajectories from terminal states is not applicable. The way to circumvent this singularity is to solve the optimal trajectories of a two-player game between the intruder and the closer defender, and assemble them with the trajectory of the other defender. The key contribution of this paper is the solution of the intruder-closer-defender subgame against a circular target area. In the vector field of the optimal trajectories, two singular surfaces and a singular point are observed. Each singular surface indicates a discontinuity in the closer defender’s control, while the singular point represents a situation where the target is successfully protected by a single defender. The complete solution of the two-defender game is solved based on the result of the intruder-closer-defender subgame. The proposed solution is verified through a special case where the capture range is zero. This verification also presents a simpler approach of solving the zero capture range problem.