Stabilization of Parabolic Nonlinear Systems with Finite Dimensional Feedback or Dynamical Controllers: Application to the Navier-Stokes System

Let $A:\mathcal{D}(A)\to\mathcal{X}$ be the generator of an analytic semigroup and $B:\mathcal{V}\to[\mathcal{D}(A^*)]'$ a relatively bounded control operator. In this paper, we consider the stabilization of the system $y'=Ay+Bu$, where $u$ is the linear combination of a family $(v_1,\ldots,v_K)$. Our main result shows that if $(A^*,B^*)$ satisfies a unique continuation property and if $K$ is greater than or equal to the maximum of the geometric multiplicities of the unstable modes of $A$, then the system is generically stabilizable with respect to the family $(v_1,\ldots,v_K)$. With the same functional framework, we also prove the stabilizability of a class of nonlinear systems when using feedback or dynamical controllers. We apply these results to stabilize the Navier-Stokes equations in two and three dimensions by using boundary controls.