Control at a distance of the motion of a rigid body immersed in a two-dimensional viscous incompressible fluid

We consider the motion of a rigid body immersed in a two-dimensional viscous incompressible fluid with Navierslip-with-friction conditions at the solid boundary. The fluid-solid system occupies the whole plane. We provethe small-time exact controllability of the position and velocity of the solid when the control takes the form of adistributed force supported in a compact subset (with nonvoid interior) of the fluid domain, away from the body.The strategy relies on the introduction of a small parameter: we consider fast and strong amplitude controls forwhich the Navier-Stokes+rigid body'' system behaves like a perturbation of the Euler+rigid body’’ system. Bythe means of a multi-scale asymptotic expansion we construct a controlled solution to the Navier-Stokes+rigidbody'' system thanks to some controlled solutions to Euler+rigid body’'-type systems and to a detailed analysis ofthe influence of the boundary layer on the solid motion.