  We present a guidance by repulsion model based on a driver-evader interaction
where the driver, assumed to be faster than the evader, follows the evader but
cannot be arbitrarily close to it, and the evader tries to move away from the
driver beyond a short distance. The key ingredient allowing the driver to guide
the evader is that the driver is able to display a circumvention maneuver
around the evader, in such a way that the trajectory of the evader is modified
in the direction of the repulsion that the driver exerts on the evader. The
evader can thus be driven towards any given target or along a sufficiently
smooth path by controlling a single discrete parameter acting on driver's
behavior. The control parameter serves both to activate/deactivate the
circumvention mode and to select the clockwise/counterclockwise direction of
the circumvention maneuver. Assuming that the circumvention mode is more
expensive than the pursuit mode, and that the activation of the circumvention
mode has a high cost, we formulate an optimal control problem for the optimal
strategy to drive the evader to a given target. By means of numerical shooting
methods, we find the optimal open-loop control which reduces the number of
activations of the circumvention mode to one and which minimizes the time spent
in the active~mode. Our numerical simulations show that the system is highly
sensitive to small variations of the control function, and that the cost
function has a nonlinear regime which contributes to the complexity of the
behavior of the system, so that a general open-loop control would not be of
practical interest. We then propose a feedback control law that corrects from
deviations while preventing from an excesive use of the circumvention mode,
finding numerically that the feedback law significantly reduces the cost
obtained with the open-loop control.
