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.
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