Asymptotic behavior and control of a guidance by repulsion model

We model and analyze a guiding problem, where the drivers try to steer the evaders' positions toward a target region while the evaders always try to escape from drivers. This problem is motivated by the guidance-by-repulsion model [R. Escobedo, A. Ibañez and E. Zuazua, Optimal strategies for driving a mobile agent in a guidance by repulsion model, Commun. Nonlinear Sci. Numer. Simul. 39 (2016) 58-72] where the authors answer how to control the evader's position and what is the optimal maneuver of the driver. First, we analyze well posedness and behavior of the one-driver and one-evader model, assuming of the same friction coefficients. From the long-time behavior, the exact controllability is proved in a long enough time horizon. Then, we extend the model to the multi-driver and multi-evader case. We assumed three interaction rules in the context of collective behavior models: flocking between evaders, collision avoidance between drivers and repulsive forces between drivers and evaders. These interactions depend on the relative distances, and each agent is assumed to be undistinguishable and obtained an averaged effect from the other individuals. In this model, we develop numerical simulations to systematically explore the nature of controlled dynamics in various scenarios. The optimal strategies turn out to share a common pattern to the one-driver and one-evader case: the drivers rapidly occupy the position behind the target, and want to pursuit evaders in a straight line for most of the time. Inspired by this, we build a feedback strategy which stabilizes the direction of evaders.