Cooperative Pursuit-Evasion Games with a Flat Sphere Condition

In planar pursuit-evasion differential games considering a faster pursuer and a slower evader, the interception points resulting from equilibrium strategies lie on the Apollonius circle. This property is instrumental for leveraging geometric approaches for solving multiple pursuit-evasion scenarios in the plane. In this paper, we study a pursuit-evasion differential game on a sphere and generalize the planar Apollonius circle to the spherical domain. For the differential game, we provide equilibrium strategies for all initial positions of the pursuer and evader, including a special case when they are on the opposite sides of the sphere and on the same line with the center of the sphere when there are infinitely many geodesics between the two players. In contrast to planar scenarios, on the sphere we find that the interception point from the equilibrium strategies can leave the Apollonius domain boundary. We present a condition to ensure the intercept point remains on the boundary of the Apollonius domain. This condition allows for generalizing planar pursuit-evasion strategies to the sphere, and we show how these results are applied by analyzing the scenarios of target guarding and two-pursuer, single evader differential games on the sphere.