Numerical method for Nash equilibrium strategies of spacecraft orbit pursuit-evasion game based on continuous thrust reachable domain analysis

A numerical method for computing Nash equilibrium strategies (NES) of the spacecraft time-optimal orbit pursuit-evasion game (TOOPEG) with continuous thrust reachable domain (RD) analysis is proposed. Through theoretical derivation and Monte Carlo validation, the equivalence among the minimum time of the TOOPEG problem with NES, the minimum time of a virtual single spacecraft for a time-optimal approach to the origin, and the minimum time required for the envelope of the pursuer’s RD to enclose that of the evader is established. First, the necessary conditions for NES are derived using Pontryagin’s maximum principle (PMP), converting the original bilateral optimal control problem into a 7-dimensional two-point boundary value problem (TPBVP). Then, the TOOPEG is transformed into a virtual single-spacecraft time-optimal approach problem, with the above necessary conditions. By exploiting the evolutionary characteristics of the continuous-thrust RD, the problem is further reduced to a 3-dimensional nonlinear differential equation. An improved Broyden quasi-Newton iterative (IBQNI) algorithm is employed to obtain high-precision numerical solutions, and an iterative initial value construction method based on a linearized orbit dynamic model is proposed. Furthermore, a set of criteria is developed to assess the relative spatial configuration between the RD of different spacecraft. Numerical simulations demonstrate that the proposed method achieves excellent convergence and remarkable computational efficiency.