Planar Herding of Multiple Evaders by a Single Pursuer

In this paper, a planar herding problem is addressed, where a single superior pursuer herds a flock of non-cooperative, inferior evaders around a predefined target point. An inverse square law of repulsion is assumed between the pursuer and each evader. It is demonstrated that a constant-velocity, circular trajectory of the pursuer, encompassing all the evaders and centered around the target point, guarantees the herding of the evaders into an arbitrarily small limit cycle around the target point. The conditions for the stability of this limit cycle, as well as the radius of the limiting herd, are derived as functions of the pursuer’s radius, angular velocity, and the strength of pursuer-evader repulsion. Estimates of the region of attraction for this stable limit cycle are computed and are found to lie within a larger unstable limit cycle, which is itself contained within the pursuer’s trajectory.