Numerical integrqation of the differential inclusion $$\dot x \in v(x,\xi(t)) + B(0,1).$$ In these movies, \(x\) is in the plane. The blue region is the reachable set, the red dot(s) display the posiyion of \(\xi\). The task of \(\xi\) is to confine \(x\) within a given prescribed region. To fix the ideas, thinl at \(x(t)\) as being the positions of sheep in a herd and \(\xi(t)\) the position(s) of shepherd dog(s).
Movie 1: sheeps are confined.
Movie 2: the dog is too slow and the sheep eventually wander around all plane.
Movie 3: the dog is able to push the whole herd from a place to another one.
Movie 4: two dogs confine the sheep.
Movie 5: the reachable set may grow exponentially.