Efficient and Robust Area Capture for Multispacecraft Orbital Games Using Hopf Formula

The spacecraft pursuit-evasion games are crucial for enabling autonomous decision-making capabilities in non-cooperative rendezvous and proximity operation scenarios. However, the lack of scalable methods significantly impedes the practical development of multi-spacecraft games. In this paper, we address the multi-spacecraft pursuit-evasion problem with area capture, a generalization of the intercept and rendezvous games based on point capture. First, the qualitative solution of the game is characterized by the backward reachability set derived from the zero-level set of the value function of the Hamilton-Jacobi-Isaacs partial differential equations (HJI PDE). The explicit Hopf formula is used to address the curse of dimensionality inherent in dynamic programming. Then, the minimum time feedback strategy is efficiently constructed by solving a series of unconstrained convex subproblems, ensuring a win within a finite time. A large disturbance compensation method is proposed to depict the degradation of capture time through an augmented evader. The game of kind and the game of degree are unified in the HJI framework. The strategy synthesis of multi-pursuer multi-evader is also conducted using dynamical maximum graph matching to reduce the terminal time of group capture. Finally, extensive numerical simulations demonstrate the effectiveness and efficiency of the proposed strategies, and the robustness and performance guaranteed are analyzed. The strong time consistency of the proposed strategy is validated in nonlinear dynamics with measurement uncertainty.