HJI-Free Solutions to Isaacs Pursuit-Evasion Games

Continuous pursuit-evasion differential games, originally formulated by Rufus Isaacs, represent the fundamental mathematical framework governing competitive autonomous interactions between adversarial dynamic agents. For decades, the aerospace and control theory communities have operated under the central premise that determining optimal feedback strategies in zero-sum competitive environments strictly necessitates solving the non-linear Hamilton-Jacobi-Isaacs partial differential equation. However, real-time onboard execution has remained persistently hindered by the curse of dimensionality, as grid-based numerical level-set toolboxes demand intractable memory allocations that prevent scaling beyond isolated single-adversary scenarios. Conversely, simplified reactive approximations, such as classical artificial potential fields and univariate avoidance heuristics, fail catastrophically under synchronized multi-axis attacks, suffering from internal saddle traps or fatal steering into orthogonal intercept trajectories. In this work, we demonstrate that optimal multi-threat pursuit-evasion does not require integrating the non-linear Hamilton-Jacobi-Isaacs partial differential equation. We introduce an entirely HJI-free geometric framework that reformulates multi-threat evasion into an overdetermined affine constraint problem projected directly onto the transverse collision manifold. By establishing the existence of a unique minimum-norm projection that simultaneously satisfies all active transverse safety boundaries, we prove that an evader can execute guaranteed trap-free escape maneuvers that resolve coordinated crossfire without computing continuous value functions, iterative game trees, or discretized reachability sets. This paradigm completely decouples adversarial evasion from numerical differential equation integration, providing instantaneous deterministic decision-making on low-power embedded processors while preserving the core geometric safety boundaries of continuous game theory. The resulting formulation bridges the historical divide between theoretical differential games and real-time autonomous vehicle survivability.