Minimum-Time Escape from the Conic Field of View of a Reeds-Shepp Pursuer: Classification and Game Value for a Slower or Equally Fast Evader

Abstract

We study minimum-time escape of an omnidirectional evader from the forward conic field of view of a Reeds-Shepp pursuer with minimum turning radius $r$. For speed ratio $ν$ and cone half-angle $0<α<π/2$, we prove a global escape/holding classification: straight reversal holds the whole cone for $ν\le\sinα$, escape is confined to a barrier-bounded dart for $\sinα<ν\le1$, and escape is possible from every state for $ν>1$. For the slower or equally fast evader, $\sinα<ν\le1$, we construct the solution with Isaacs' characteristics and verify it in feedback form, using a Filippov pursuer relay and a one-sided evader branch. The finite escape time is the first root of one scalar equation, and the holding region is bounded by the nonusable parts of the rays and a barrier that closes with a convex corner on the front axis. Sample-and-hold counter-controls show that this escape time is the value of the game in nonanticipative strategies at every state except the corner. There the two strategy values differ: the pursuer's is infinite, whereas an evader that chooses which barrier branch to cross from the initial germ of the pursuer's control escapes before the barrier time $τ_I$, which is the evader's exact strategy value. An independent level-set solver, refined on fixed comparison points, agrees with the closed-form escape time and reproduces the holding region at the tested parameters.