A Dimension-reduction method for the finite-horizon spacecraft pursuit-evasion game

The finite-horizon two-person zero-sum differential game is a significant technology to solve the finite-horizon spacecraft pursuit-evasion game (SPE game). Considering that the saddle point solution of the differential game usually results in solving a high-dimensional (24 dimensional in this paper) two-point boundary value problem (TPBVP) that is challengeable, a dimension-reduction method is proposed in this paper to simplify the solution of the 24-dimensional TPBVP related to the finite-horizon SPE game and to improve the efficiency of the saddle point solution. In this method, firstly, the 24-dimensional TPBVP can be simplified to a 12-dimensional TPBVP by using the linearization of the spacecraft dynamics; then the adjoint variables associated with the relative state variables between the pursuer and evader can be expressed in the form of state transition; after that, based on the necessary conditions for the saddle point solution and the adjoint variables in the form of state transition, the 12-dimensional TPBVP can be transformed into the solving of 6-dimensional nonlinear equations; finally, a hybrid numerical algorithm is developed to solve the nonlinear equations so as to obtain the saddle point solution. The simulation results show that the proposed method can effectively obtain the saddle point solution and is robust to the interception time, the orbital altitude and the initial relative states between the pursuer and evader.