Interior and Boundary Stabilization of Navier-Stokes Equations

We report on very recent work on the stabilization of the steady-state solutions to Navier-Stokes equations on an open bounded domain R C Rd, d = 2,3, by either interior, or else boundary control. More precisely, as to the interior case, we obtain that the steadystate solutions to Navier-Stokes equations on R C Rd, d = 2,3, with no-slip boundary conditions, are locally exponentially stabilizable by a finite-dimensional feedback controller with support in an arbitrary open subset w C R of positive measure. The (finite) dimension of the feedback controller is minimal and is related to the largest algebraic multiplicity of the unstable eigenvalues of the linearized equation. Second, as to the boundary case, we obtain that the steady-state solutions to Navier-Stokes equations on a bounded domain R C Rd, d = 2,3, are locally exponentially stabilizable by a boundary closedloop feedback controller, acting on the boundary 80, in the Dirichlet boundary conditions. If d = 3, the non-linearity imposes and dictates the requirement that stabilization must occur in the space (H;+'(R))~, E > 0, a high topological level. A first implication thereof is that, for d = 3, the boundary feedback stabilizing controller must be infinite dimensional. Moreover, it generally acts on the entire boundary dR. Instead, for d = 2, where the topological level for stabilization is (H$-"(R))', 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.