  The present paper provides a solution in the affirmative to a recognized open
problem in the theory of uniform stabilization of 3-dimensional Navier-Stokes
equations in the vicinity of an unstable equilibrium solution, by means of a
`minimal' and `least' invasive feedback strategy which consists of a control
pair $\{ v,u \}$ \cite{LT2:2015}. Here $v$ is a tangential boundary feedback
control, acting on an arbitrary small part $\widetilde{\Gamma}$ of the boundary
$\Gamma$; while $u$ is a localized, interior feedback control, acting
tangentially on an arbitrarily small subset $\omega$ of the interior supported
by $\widetilde{\Gamma}$. The ideal strategy of taking $u = 0$ on $\omega$ is
not sufficient. A question left open in the literature was: Can such feedback
control $v$ of the pair $\{ v, u \}$ be asserted to be finite dimensional also
in the dimension $d = 3$? We here give an affirmative answer to this question,
thus establishing an optimal result. To achieve the desired finite
dimensionality of the feedback tangential boundary control $v$, it is here then
necessary to abandon the Hilbert setting of past literature and replace it with
a Besov setting which are `close' to $L^3(\Omega)$ for $d=3$. It is in line
with recent critical well-posedness in the full space of the non-controlled N-S
equations. A key feature of such Besov spaces with tight indices is that they
do not recognize compatibility conditions. The proof is constructive and is
"optimal" also regarding the "minimal" number of tangential boundary
controllers needed. The new setting requires establishing maximal regularity in
the required critical Besov setting for the overall closed-loop linearized
problem with tangential feedback control applied on the boundary. Finally, the
minimal amount of tangential boundary action is linked to the issue of unique
continuation of over-determined Oseen eigenproblems.
