Locally Optimizing Strategies for Approaching the Furthest Evader

We describe a method for constructing feedback strategies based on minimizing/maximizing state evaluation functions with use of steepest descent/ascent conditions. For a specific kinematics, not all control variables may be presented implicitly in the corresponding optimality conditions and some additional local conditions are to be invoked to design strategies for these controls. We apply the general technics to evaluate a chance for the pursuer $P$ to approach the real target that uses decoys by a kill radius $r$. Assumed that $P$ cannot classify the real and false targets. Therefore, $P$ tries to come close to the furthest evader, and thereby to guarantee the capture of all targets including the real one. We setup two-person zero-sum differential games of degree with perfect information of the pursuing, $P$, and several identical evading, $E_1,\ldots,E_N$, agents. The $P$'s goal is to approach the furthest of $E_1,\ldots,E_N$ as closely as possible. Euclidean distances to the furthest evader at the current state or their smooth approximations are used as evaluation functions. For an agent with simple motion, the method allows to specify the strategy for heading angle completely. For an agent that drives a Dubins or Reeds-Shepp car, first we define his targeted trajectory as one that corresponds to the game where all agents has simple motions and apply the locally optimal strategies for heading angles. Then, to design strategies for angular and ordinary velocities, local conditions under which the resulting trajectories approximate the targeted ones are invoked. Two numerical examples of the pursuit simulation when one or two decoys launched at the initial instant are given.