Exact boundary controllability of Galerkin's approximations of Navier-Stokes equations

We consider the 2-d and 3-d Navier - Stokes equations in a bounded smooth domain with a boundary control acting on the system through the Navier slip boundary conditions. We introduce a finite-dimensional Galerkin approximation of this system. Under suitable assumptions on the Galerkin basis we prove that this Galerkin approximation is exactly controllable. Moreover we prove that the cost of controlling is independent of the presence of the nonlinearity on the system. Our assumptions on the Galerkin basis are related to the linear independence of suitable traces of its elements over the boundary. At this respect, the one-dimensional Burgers equation provides a particularly degenerate example that we study in detail. In this case we prove local controllability results.