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

TL;DR

This paper proves that the 2-D Vlasov-Stokes system on a torus can be exactly controlled in small time using internal controls, allowing transition between small initial and final states.

Contribution

It establishes the first exact controllability result for the 2-D Vlasov-Stokes system with small data using the return method and fixed-point techniques.

Findings

Exact controllability in small time for small data

Control can steer initial states to desired final states

Method applicable to 2-D Vlasov-Stokes system on a torus

Abstract

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 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.