Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control

Let -A: D (A) H be the generator of an analytic semigroup and B: U D (A^*) ' a relatively bounded control operator such that (A-, B) is stabilizable for some >0. In this paper, we consider the stabilization of the nonlinear system y'+Ay+G (y, u) =Bu by means of a feedback or a dynamical control u. The control is obtained from the solution to a Riccati equation which is related to a low-gain optimal quadratic minimization problem. We provide a general abstract framework to define exponentially stable solutions which is based on the contruction of Lyapunov functions. We apply such a theory to stabilize, around an unstable stationary solution, the 2D or 3D Navier-Stokes equations with a Neumann control and the 2D or 3D Boussinesq equations with a Dirichlet control.