STABILIZATION OF PARABOLIC NONLINEAR SYSTEMS WITH FINITE DIMENSIONAL FEEDBACK OR DYNAMICAL CONTROLLERS: APPLICATION TO THE NAVIER-STOKES SYSTEM

摘要

Let A : D(A) -> X be the generator of an analytic semigroup and B : V -> [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), ..., 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), ..., 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.