  We consider 2- or 3-dimensional incompressible Navier-Stokes equations
defined on a bounded domain $\Omega$, with no-slip boundary conditions and
subject to an external force, assumed to cause instability. We then seek to
uniformly stabilize such N-S system, in the vicinity of an unstable equilibrium
solution, in critical $L^q$-based Sobolev and Besov spaces, by finite
dimensional feedback controls. These spaces are `close' to $L^3(\Omega)$ for
$d=3$. This functional setting is significant. In fact, in the case of the
uncontrolled N-S dynamics, extensive research efforts have recently lead to the
space $L^3(\mathbb{R}^3)$ as being a critical space for the issue of
well-posedness in the full space. Thus, our present work manages to solve the
stated uniform stabilization problem for the controlled N-S dynamics in a
correspondingly related function space setting. In this paper, the feedback
controls are localized on an arbitrarily small open interior subdomain $\omega$
of $\Omega$. In addition to providing a solution of the uniform stabilization
problem in such critical function space setting, this paper manages also to
much improve and simplify, at both the conceptual and computational level, the
solution given in the more restrictive Hilbert space setting in the literature.
Moreover, such treatment sets the foundation for the authors' final goal in a
subsequent paper. Based critically on said low functional level where
compatibility conditions are not recognized, the subsequent paper solves in the
affirmative a presently open problem: whether uniform stabilization by
localized tangential boundary feedback controls, which-in addition-are finite
dimensional, is also possible in dim $\Omega = 3$.
