  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.

Optimization and Control