ABSTRACT

We study here stability problem for two-dimensional (2D) Navier-Stokes equations defined in a bounded domain These equations are controlled by Dirichlet boundary condition for velocity vector field. Let be a steady-state solution for Navier-Stokes equations. Suppose that is an unstable singular point for the dynamic system generated by evolutionary Navier-Stokes system supplied with zero condition v| ∂Ω=0 on the boundary ∂Ω of Ω. The stabilization problem is as follows: Given σ>0 and initial condition v 0 (x) for evolutionary Navier-Stokes system (v 0 is placed in a small H

find Dirichlet boundary condition (control) u(t, x′) defined on such that the solution v(t,x) of obtained boundary value problem satisfies inequality

Actually, the stabilization result in such formulation follows immediately from exact controllability result of [FE], [F1]. But below we impose on desired control

very important additional property: u should be feedback control, i.e. u must react on

(1.1)

unpredictable fluctuations of v suppressing them. Problem of stabilization by boundary feedback control was studied earlier mostly for

hyperbolic equations and related to them systems (see, for instance, [Li], [Lag], [C]). Some results, connected with Burgers equation was obtained also ([BK]). (Of course, we do not pretend on completeness of references.)

In [F2] new mathematical formalization of feedback property was proposed and stabilization problem for quasilinear parabolic equation with feedback boundary control was solved. Here we get solution for problem of stabilization by boundary feedback control for two-dimensional Navier-Stokes system when a control is concentrated on a part of boundary. Complete proof of this result will be exposed in [F3].