Interior and Boundary Stabilization of Navier-Stokes Equations - Roberto Triggiani (Boletim da Sociedade Paranaense de Matemática, 2004)
We report on very recent work on the stabilization of the steady-state solutions to Navier-Stokes equations on an open bounded domain Omega subset R^d, d = 2; 3, by either interior, or else boundary control. More precisely, as to the interior case, we obtain that the steady-state solutions to Navier-Stokes equations on Omega subset R^d, 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 omega subset Omega of positive measure. The (finite) dimension of the feedback controller is minimal and is related to the largest algebraicmultiplicity 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 Omega subset R^d, d = 2; 3, are locally exponentially stabilizable by a boundary closed-loop feedback controller, acting on the boundary partial Omega, in the Dirichlet boundary conditions. If d = 3, the non-linearity imposes and dictates the requirement that stabilization must occur in the space H^{3/2+epsilon}(Omega))^3, epsilon > 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 @. Instead, for d = 2, where thetopological level for stabilization is (H^{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.