Stabilizability of Two-Dimensional Navier—Stokes Equations with Help of a Boundary Feedback Control

Abstract.

For 2D Navier—Stokes equations defined in a bounded domain $\Omega$ we study stabilization of solution near a given steady-state flow $\hat v(x)$ by means of feedback control defined on a part $\Gamma$ of boundary $\partial\Omega$. New mathematical formalization of feedback notion is proposed. With its help for a prescribed number $\sigma > 0$ and for an initial condition v ₀(x) placed in a small neighbourhood of $\hat v(x)$ a control u(t,x'), $x' \in \Gamma$, is constructed such that solution v(t,x) of obtained boundary value problem for 2D Navier—Stokes equations satisfies the inequality: $\|v(t,\cdot)-\hat v\|_{H^1}\leqslant ce^{-\sigma t}\quad {\rm for}\; t \geqslant 0$. To prove this result we firstly obtain analogous result on stabilization for 2D Oseen equations.