Optimal Interception of Evasive Missile Warheads: Numerical Solution of the Differential Game

Abstract

The problem of interception of a ballistic missile warhead by a defending missile is formulated as a differential game. Each missile is given a modest postlaunch capability to maneuver. Interception concludes the game and occurs if the interceptor reduces the distance between it and the warhead to a specified value. The objective of the warhead is to minimize the final distance to the target, which lies on the Earth's surface but is not necessarily collocated with the interceptor launch site. The objective of the interceptor is to maximize this same distance at the capture time. Saddle-point equilibrium solutions are found using a recently developed, direct numerical method that nevertheless uses the analytical necessary conditions to find the optimal control for one of the players. The method requires an initial guess of the solution, and this is provided by generating an approximate solution using genetic algorithms. For initial conditions yielding eventual capture, we derive the necessary condition that the saddle-point trajectories terminate on the usable part of the terminal hypersurface; this condition is shown to have an interesting physical interpretation. The numerical method successfully finds the saddle-point trajectories and is used to determine the sensitivity of the value of the game to the capability of the interceptor, for design purposes, and also to gauge the robustness of the numerical method for the solution of this dynamic game.