Exact controllability and feedback stabilization from a boundary for the navier-stokes equations

Abstract

For 2D Navier-Stokes equations defined in a bounded domain Ω we study stabilization of solution near a given steady-state flow v̂(x) by means of feedback control defined on a part Γ of boundary ∂Ω. New mathematical formalization of feedback notion is proposed. With its help for a prescribed number σ > 0 and for an initial condition v 0(x) placed in a small neighbourhood of v̂(x) a control u(t, x′), x ∈ Γ, is constructed such that solution v(t, x) of obtained boundary value problem for 2D Navier-Stokes equations satisfies the inequality: ||v(t,·)-v̂||h1 ≤ ce-σt for t ≥ 0. © 2006 Springer-Verlag Berlin Heidelberg.