Constrained Pursuit - Evasion Problems in the Plane

In pursuit-evasion problems, we are presented with one or more pursuers attempting to capture one or more evaders. We consider pursuers and evaders limited by a maximum speed moving in the two-dimensional plane with obstacles. We then investigate two problems in this domain. In the first, where we are given the starting configuration of pursuers and evaders, we identify all possible paths by the evaders that are not intercepted by pursuers, and the points reachable by the evaders before the pursuers by following these paths. In the second problem, we consider a pursuer forced to maintain visibility with an evader. We construct an example that demonstrates there exists, in addition to the two standard outcomes of the pursuer capturing the evader and the evader losing sight of the pursuer, a third tie outcome, where the pursuer never loses sight of the evader, but the evader can also avoid capture indefinitely. We give the conditions under which each of these three outcomes occur for our specific situation.