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This paper investigates the proximity satellite pursuit-evasion game where the pursuer carries three orthogonal thrusters, each of which can exert a magnitude-limited force along one axis, while the evader has a single thruster which can produce a bounded acceleration in any direction. The pursuer or the evader tries their best to capture or escape by performing orbital maneuvers. By establishing a local moving coordinate frame on the originally revolving orbit of the evader, we reduce the dynamics of each player to the linear Clohessy-Wiltshire equations. We then transform the problem of finding the optimal controls associated with the saddle point solution of the game into the two-point boundary value problem, which is solved by combining the heuristic searching and Newton method. At last, by numerical simulations, we discuss the effectiveness of the proposed algorithm in finding the open-loop solution to the game. We show that, in contrast with pursuit-evasion games where each player has one single thruster, the problem considered in this paper may not be solved efficiently by the indirect method, since there exist some initial states of the players such that the proposed algorithm fails to solve the open-loop control.

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