The steady-state solutions to Navier-Stokes equations on a
bounded domain $\Omega \subset R^d$, $d = 2,3$, are locally
exponentially stabilizable by a boundary closed-loop feedback controller,
acting tangentially on the boundary $\partial \Omega$, in the
Dirichlet boundary conditions. The greatest challenge arises from a
combination between the control as acting on the boundary and the
dimensionality $d=3$. If $d=3$, the non-linearity imposes and
dictates the requirement that stabilization must occur in the space
$(H^{\tfrac{3}{2}+\epsilon}(\Omega))^3$, $\epsilon > 0$, a
high topological level. A first implication thereof is that, due to
compatibility conditions that now come into play, for $d=3$, the
boundary feedback stabilizing controller *must* be infinite
dimensional. Moreover, it generally acts on the entire boundary $\partial
\Omega$. Instead, for $d=2$, where the topological level for
stabilization is $(H^{\tfrac{3}{2}-\epsilon}(\Omega))^2$, the boundary
feedback stabilizing controller can be chosen to act on *an arbitrarily
small* portion of the boundary. Moreover, still for $d=2$, it may
even be *finite* dimensional, and this occurs if the linearized
operator is diagonalizable over its finite-dimensional unstable subspace.

In order to inject dissipation as to force local exponential
stabilization of the steady-state solutions, an Optimal Control Problem (OCP)
with a quadratic cost functional over an infinite time-horizon is introduced
for the linearized N-S equations. As a result, the same Riccati-based, optimal
boundary feedback controller which is obtained in the *linearized* OCP
is then selected and implemented also on the full N-S system. For
$d=3$, the OCP falls definitely outside the boundaries of established
optimal control theory for parabolic systems with boundary controls, in that
the combined index of unboundedness—between the unboundedness of the
boundary control operator and the unboundedness of the penalization or
observation operator—is *strictly larger than*
$\tfrac{3}{2}$, as expressed in terms of fractional powers of the
free-dynamics operator. In contrast, established (and rich) optimal control
theory [L-T.2] of boundary control parabolic problems and corresponding
algebraic Riccati theory requires a combined index of unboundedness
*strictly less than* 1. An additional preliminary serious difficulty to
overcome lies at the outset of the program, in establishing that the present
highly non-standard OCP—with the aforementioned high level of
unboundedness in control and observation operators and subject, moreover, to
the additional constraint that the controllers be *pointwise
tangential*—be non-empty; that is, it satisfies the so-called Finite
Cost Condition [L-T.2].