A simple pursuit game in positions partitioned by an obstacle

With necessary optimality conditions a simple game is studied of pursuit in a plane with a circle cutoff for the case where the stretch connecting the positions of players has at least one common point with the obstacle. Strategies of geodetic pursuit and evasion are considered which assign the movement for the pursuer and the evader along the geodetic line and its extension. Assuming that the pursuer uses the geodetic pursuit strategy a range of positions is identified where the geodetic evasion strategy is not optimal. The structure of optimal paths for positions in that range is analyzed.