  We consider the 2D incompressible Navier-Stokes equation in a rectangle with
the usual no-slip boundary condition prescribed on the upper and lower
boundaries. We prove that for any positive time, for any finite energy initial
data, there exist controls on the left and right boundaries and a distributed
force, which can be chosen arbitrarily small in any Sobolev norm in space, such
that the corresponding solution is at rest at the given final time.
  Our work improves earlier results where the distributed force is small only
in a negative Sobolev space. It is a further step towards an answer to
Jacques-Louis Lions' question about the small-time global exact boundary
controllability of the Navier-Stokes equation with the no-slip boundary
condition, for which no distributed force is allowed.
  Our analysis relies on the well-prepared dissipation method already used for
Burgers and for Navier-Stokes in the case of the Navier slip-with-friction
boundary condition. In order to handle the larger boundary layers associated
with the no-slip boundary condition, we perform a preliminary regularization
into analytic functions with arbitrarily large analytic radius and prove a
long-time nonlinear Cauchy-Kovalevskaya estimate relying only on horizontal
analyticity.
