On the controllability of the Vlasov–Poisson system

In this paper, we study the controllability of the Vlasov–Poisson system in a periodic domain, by means of an interior control located in an spatial subdomain. The first result proves the local exact zero controllability in the two-dimensional torus between two small acceptable distribution functions, with an arbitrary control zone. A second result establishes the global exact controllability in arbitrary dimension, provided the control zone satisfies the condition that it contains a hyperplane of the torus.