Stabilization of Solutions to Navier-Stokes System by Feedback Boundary Control

 

Abstract

For 2D or 3D 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 neighborhood 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: 
$\|v(t,\cdot)-\hat v\|_{H^1}\le ce^{-\sigma t}\quad \mbox{for}\;  t\ge  0.$
To  prove this result we firstly obtain analogous result on stabilization for 2D Oseen equations.

Besides,we introduce a notion of real process that is an abstract analog of approximate solution obtained as a result of numerical calculations. Investigation a structure of real process gives us a possibility to construct feedback stabilization from the boundary $\partial \Omega$ for it and to obtain an estimate for stabilized real process. The result of these investigation is construction of feedback control from the boundary  such that it stabilizes solution of 3D Navier-Stokes equations. Moreover, it can react on unpredictable fluctuations of solution when they arise, damping them.