In this paper, we prove a controllability
result for a fluid-structure interaction problem. In dimension two,
a rigid structure moves into an incompressible fluid governed by
Navier-Stokes equations. The control acts on a fixed subset of the
fluid domain. We prove that, for small initial data, this system is
null controllable, that is, for a given T > 0, the system can be
driven at rest and the structure to its reference configuration at
time T. To show this result, we first consider a linearized
system. Thanks to an observability inequality obtained from a
Carleman inequality, we prove an optimal controllability result with
a regular control. Next, with the help of Kakutani's fixed point
theorem and a regularity result, we pass to the nonlinear problem.