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

For 2D Navier-Stokes equations defined in a bounded domain Omega we study stabilization of solution near a given steady-state flow (v) over cap (x) by means of feedback control defined on a part Gamma of boundary partial derivative 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 (v) over cap (x) a control u(t, x'), x' is an element of Gamma, is constructed such that solution v(t, x) of obtained boundary value problem for 2D Navier-Stokes equations satisfies the inequality: parallel to v(t, .) - (v) over cap parallel to(H1) <= ce-(sigma t) for t >= 0. To prove this result we firstly obtain analogous result on stabilization for 2D Oseen equations.