On the terminal conditions of the two cutters and a fugitive ship differential game with non-zero capture radius and different players’ speed ratio

We study a pursuit-evasion differential game in which three agents move with simple motion on the Euclidean plane. Two of them, the cutters (pursuers), aim to capture a fugitive ship (the evader) as soon as possible. Having an opposite goal, the fugitive ship seeks to avoid capture for as long as possible. The game ends when the distance between the fugitive and at least one of the cutters is smaller than a given value. We have divided the game into two cases: case 1, when all players have the same speed, and case 2, when the evader is faster than the pursuers. Unlike previous work, our main innovations are as follows. For case 1, we present a solution obtained using exclusively differential game techniques. The game of kind is solved by establishing a study of the barrier that defines the winner of the game. In addition, we obtained time-optimal strategies for all players, proving that they do not switch controls. For case 2, we obtain the primary solution and exhibit an example showing the existence of control switches.