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

International audienceLet $A : \mathcal{D}(A)\to \mathcal{X}$ be the generator of an analytic semigroup and $B : \mathcal{U} \to [{\cal D}(A^*)]'$ a quasi-bounded 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 or equal to the maximum of the geometric multiplicities of the 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 system when using feedback or dynamical controllers. We apply these results to stabilize the Navier--Stokes equations in 2D and in 3D by using boundary control