  We model and analyze a herding problem, where the drivers try to steer the
evaders' trajectories while the evaders always move away from the drivers. This
problem is motivated by the guidance-by-repulsion model [Escobedo, R.,
Iba\~nez, A. and Zuazua, E. COMMUN NONLINEAR SCI 39 (2016) 58-72], where the
authors answer how to control the evaders' positions and what is the optimal
maneuver of the drivers. First, we obtain the well-posedness and the long-time
behavior of the one-driver and one-evader model, assuming of the same friction
coefficients. In this case, the exact controllability is proved in a long
enough time horizon. We extend the model to the multi-driver and multi-evader
case, and 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.
