Lyapunov Function and Local Feedback Boundary Stabilization of the Navier--Stokes Equations

We study the local exponential stabilization, near a given steady-state flow, of solutions of the Navier-Stokes equations in a bounded domain. The control is performed through a Dirichlet boundary condition. We apply a linear feedback controller, provided by a well-posed infinite dimensional Riccati equation. We give a characterization of the domain of the closed-loop operator which is obtained from the closed-loop linearized Navier-Stokes system. We give a class of initial data for which a Lyapunov function is obtained. For all s ∈ [0, 1/2[, the stabilization of the 2-D Navier-Stokes equations is proved for initial data in H(Ω) ∩ V 0 n (Ω), where V 0 n (Ω) is the space in which the Stokes operator is defined. We also obtain a 3-D stabilization result but only for a very specific set of initial data.