  In this paper, we investigate a pursuit-evasion game in which a mobile
observer tries to track a target in an environment containing obstacles. We
formulate the game as an optimal control problem with state inequality
constraint in a simple environment. We show that for some initial conditions,
there are two different regimes in the optimal strategy of the pursuer
depending on whether the state-constraint is activated. We derive the equations
that characterize the switching time between the two regimes. The pursuer's
optimal tracking strategy in a simple environment is further extended to a
general environment with multiple polygonal obstacles. We propose techniques to
construct a "pursuit field" based on the optimal solutions to guide the motion
of the observer in a general environment.
