Evasion differential game with coordinate-wise integral constraints on controls of players

We study an evasion differential game of one evader against one pursuer in the plane R2 with coordinate-wise integral constraints on the control functions of players. The game is described by some differential equations in terms of each coordinate. The evader moves within a small neighborhood of a vertical 0-axis, either by moving vertically or maneuvering. The evader maneuvers to the right, if the pursuer is on its left side and vice versa. We say that evasion is possible, if the position of the evader does not coincide with that of the pursuer at all times. We obtain a sufficient condition of evasion and construct an explicit strategy for the evader to ensure evasion. The strategies depend on the initial positions of players and a defined approached distance between the pursuer and the evader. Each strategy is shown to be admissible by using the fact that the integral constraints are coordinate-wise. By these strategies, evasion is proved to be possible from any initial position of players.

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