On the controllability of the 2-D Vlasov–Stokes system

In this paper we prove an exact controllability result for the Vlasov–Stokes system in the two-dimensional torus with small data by means of an internal control. We show that one can steer, in arbitrarily small time, any initial datum of class $\mathscr{C}^1$ satisfying a smallness condition in certain weighted spaces to any final state satisfying the same conditions. The proof of the main result is achieved thanks to the return method and a Leray–Schauder fixed-point argument.

Keywords

Vlasov–Stokes system, kinetic-theory, kinetic-fluid model, controllability, return method

2010 Mathematics Subject Classification

35R01, 82C40, 93C20

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Published 24 February 2017