In this paper, we present a game theoretic analysis of a visibility based pursuit-evasion game in a planar environment containing obstacles. The pursuer and the evader are holonomic having bounded speeds. Both the players have a complete map of the environment. Both the players have omnidirectional vision and have knowledge about each other's current position as long as they are visible to each other. The pursuer wants to maintain visibility of the evader for maximum possible time and the evader wants to escape the pursuer's sight as soon as possible. Under this information structure, we present necessary and sufficient conditions for surveillance and escape. We present strategies for the players that are in Nash Equilibrium. The strategies are a function of the value of the game. Using these strategies, we construct a value function by integrating the adjoint equations backward in time from the termination situations provided by the corners in the environment. From these value functions we recompute the control strategies for the players to obtain optimal trajectories for the players near the termination situation. As far as we know, this is the first work that presents the necessary and sufficient conditions for tracking for a visibility based pursuit-evasion game and presents the equilibrium strategies for the players.
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