# Confinement in Individuals-Continuum Interactions

### Patterns and Simulations

By R. M. Colombo and N. Pogodaev
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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.
 Author Institutional Affiliation Department of Mathematics, Brescia University Author Email Author Postal Mail Dipartimento di Matematica Università degli studi di Brescia Via Branze 38 25123 Brescia, Italy Keywords confinement strategies, control problems, differential inclusions