A searchlight game

Two players, P and E, not knowing each others´ positions, move in a domain. Player P has a searchlight which he can flash at will and which then illuminates a certain area around P. The game ends when E is caught within this area, provided it is illuminated. If E is not in the illuminated area, then P has disclosed his position to E since E can observe the searchlight, if it is switched on, from everywhere. P wants to maximize and E wants to minimize the capture chance over a given time horizon. The paper provides a dynamic programming formulation of this game, which in its turn yields optimal strategies for the players, i.e. how to move and for P, in addition, at which time instants to flash. The game is considered on a finite state space and in discrete time.