Multiple Pursuers Multiple Evaders Reach-Avoid Games in 3D Space via Analytic Barriers

Multiplayer differential games in three-dimensional (3D) space constitute a highly challenging yet profoundly significant research issue. This paper investigates a 3D reach-avoid differential game involving N pursuers vs M evaders, addressing both optimal assignments and optimal strategy simultaneously. Leveraging the Apollonius sphere as a fundamental tool, the evader’s safe region is characterized, and a representative measure for determining win–loss outcomes is proposed. Furthermore, the analytical expressions of the barrier are derived in one-on-one and two-on-one scenarios, which represent the first successful derivation of the analytical solution in 3D space. Subsequently, the optimal strategy guaranteeing the pursuer’s victory is formulated. Additionally, the basis for solving the optimal assignment is established utilizing the barrier measures. Numerical examples are given to illustrate the results. This paper lays the foundation for analyzing adversarial multiplayer differential games in high-dimensional spaces.