A finite difference method for calculating the closed-loop equilibrium of orbital pursuit-evasion game

When studying non-cooperative dynamic games, equilibrium is a crucial concept. Scholars typically concentrate on two types of equilibrium: open-loop and closed-loop. Currently, research in the field of orbital pursuit-evasion games primarily focuses on the construction methods of open-loop equilibrium, while studies on closed-loop equilibrium remain relatively scarce, which affects the overall understanding and in-depth comprehension of spacecraft pursuit-evasion game problems. The key to constructing closed-loop equilibrium lies in dynamic adjustment of control strategies by both players based on actual game progression. Current mainstream approaches approximate closed-loop effects by computing numerous open-loop equilibrium, but two major limitations persist: First, these methods remain constrained by initial game conditions, allowing only single-scenario solutions and hindering the extraction of generalizable patterns. Second, the computational cost for individual scenarios is prohibitively high, requiring extensive open-loop equilibrium calculations to ensure approximation accuracy. To address these challenges, this paper integrates Bellman’s optimality principle, finite difference methods, and spline interpolation techniques to propose a novel method for closed-loop equilibrium construction and control function computation. This approach enables simultaneous processing of multiple game scenarios, providing robust support for analyzing game situations and deriving universal principles.

The paper addressed the issue of constructing the closed-loop equilibrium for close-range orbital pursuit-evasion games and proposed a computation method that integrates Bellman’s principle of optimality, the finite difference method, and interpolation techniques. A dimension-reduction dynamics of the game system in the line-of-sight coordinate frame was derived, establishing a close-range orbital pursuit-evasion game model and reducing the dimensionality of the system’s state space. Based on Bellman’s principle of optimality, the original problem was reformulated as a HJI PDE terminal value problem, enabling the simultaneous handling of multiple game scenarios through reverse-time analysis. The state space was discretized using Cartesian grids, and the finite difference method was employed to calculate the dynamic evolution process of the equilibrium driven by the dynamics, and analyze the game situation. Utilizing the relationship between control and the spatial gradient of the equilibrium, numerical interpolation was applied to construct the closed-loop control function.

Numerical simulations of scenarios with different thrust configurations have demonstrated the feasibility of the proposed method; Through retrograde analysis, the impact of variable initial conditions in scenarios is eliminated. Based on the relationship between control functions and the spatial gradient of the equilibrium, closed-loop control functions are designed simultaneously for multiple scenarios to identify common patterns. This provides foundational support for analyzing game situations and summarizing general principles.

First, a reduced-dimensional dynamics model under the line-of-sight rotating coordinate system is employed to achieve state-space dimensionality reduction in the pursuit-evasion system, thereby lowering the complexity of the problem. Second, based on Bellman’s optimality principle, the HJI PDE is formulated to characterize the evolution of equilibrium solutions driven by dynamics. Retrograde analysis is applied to eliminate initial-value constraints in game scenarios, enabling simultaneous processing of multiple scenarios. Subsequently, the Cartesian grid is used to discretize the state space, and a finite difference method with a WENO-TVD solver is constructed to numerically solve the terminal-value problem of the HJI PDE, providing support for game dynamics analysis and generalization of common principles. Finally, the relationship between control and equilibrium gradients is established. Numerical interpolation is utilized to compute equilibrium and their spatial gradients under current game situations. Combined with the dynamics model, equilibrium trajectories and complete closed-loop control functions are extrapolated via numerical methods, achieving closed-loop equilibrium solutions for nonlinear pursuit-evasion problems.