Defense game in a circular region

This paper considers a particular pursuit-evasion game, called as defense game in which a faster defender patrols in a circular region and guards it against a spy. The spy aims to escape from the circular region without being point-captured, while the defender aims to prevent the escape. We are concerned with the conditions under which the defender or the spy can win the game. In this work, the stated problem is posed as a game of kind, and the central focus is the construction of the barrier that separates the capture zone from the escape zone. Since the classic Isaacs' approach cannot deal with this game which has two terminal manifolds, we propose a geometrical method to construct the barrier analytically. Then, the optimal control strategies of the players on the barrier are presented. Furthermore, when the players' initial positions lie in their winning regions, two games of degree with different payoff functions are discussed.