Two-dimensional local null controllability of a rigid structure in a Navier-Stokes fluid

We consider the two-dimensional motion of a rigid structure immersed in an incompressible fluid governed by Navier-Stokes equations. The control force acts on a fixed subset of the fluid domain. We prove that our system is null controllable; that is, for small initial data, the system can be driven at rest and the structure can be driven to the origin at a given T > 0. The result holds for a structure symmetric with respect to the center of mass and for initial conditions satisfying strong compatibility conditions