Target-Defense Games with One or Two Unmanned Surface Vehicles Defending an Island: A Geometric Analytical Approach

This paper considers a target-defense game in an open area with one or two defenders as well as an intruder. The intruder endeavors to reach the boundary of the island, while the defenders strive to prevent that by capturing the intruder through contact. Islands, as closed areas, restrict the free movement of the defenders, since the defenders—represented by USVs—cannot traverse the target area directly. First, we are concerned with the barrier, which is the boundary of the winning zones, taking into account the impact of the target. For the initial states lying in the defenders’ winning zone, there exists a strategy for the defenders to intercept the intruder regardless of the intruder’s best effort, while for the initial states lying in the intruder’s winning zone, the intruder can always invade successfully. We propose a geometric method to construct the barrier analytically for two kinds of speed ratios. Then, by taking index functions into consideration, we present optimal strategies for the players after constructing the dominance regions when their initial states lie in different winning zones. Simulation results verify the effectiveness of the proposed method. This study can be extended to scenarios involving multiple defenders.