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

Abstract

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 $s\in[0,1/2[$, the stabilization of the two-dimensional Navier-Stokes equations is proved for initial data in $\mathbf{H}^s(\Omega)\cap V_n^0(\Omega)$, where $V_n^0(\Omega)$ is the space in which the Stokes operator is defined. We also obtain a three-dimensional stabilization result but only for a very specific set of initial data.