One Apollonius Circle is Enough for Many Pursuit-Evasion Games

This paper investigates obstacle-free simple motion pursuit-evasion problems where the pursuer is faster and game termination is point capture. It is well known that the interior of the Apollonius Circle (AC) is the evader's dominance region, however, it was unclear whether the evader could reach outside the initial AC without being captured. We construct a pursuit strategy that guarantees the capture of an evader within an arbitrarily close neighborhood of the initial AC. The pursuer strategy is derived by reformulating the game into a nonlinear control problem, and the guarantee holds against any admissible evader strategy. Our result implies that the evader can freely select the capture location, but only inside the initial AC. Therefore, a class of problems, including those where the payoff is determined solely based on the location of capture, are now trivial.