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Feedback boundary stabilization of the three-dimensional incompressible Navier–Stokes equations
Authors:J-P Raymond  
Institution:aInstitut de Mathématiques de Toulouse, UMR CNRS 5219, Université Paul Sabatier, 31062 Toulouse Cedex 9, France
Abstract:We study the local stabilization of the three-dimensional Navier–Stokes equations around an unstable stationary solution w, by means of a feedback boundary control. We first determine a feedback law for the linearized system around w. Next, we show that this feedback provides a local stabilization of the Navier–Stokes equations. To deal with the nonlinear term, the solutions to the closed loop system must be in H3/2+ε,3/4+ε/2(Q), with 0<ε. In V. Barbu, I. Lasiecka, R. Triggiani, Boundary stabilization of Navier–Stokes equations, Mem. Amer. Math. Soc. 852 (2006); V. Barbu, I. Lasiecka, R. Triggiani, Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers, Nonlinear Anal. 64 (2006) 2704–2746], such a regularity is achieved with a feedback obtained by minimizing a functional involving a norm of the state variable strong enough. In that case, the feedback controller cannot be determined by a well posed Riccati equation. Here, we choose a functional involving a very weak norm of the state variable. The compatibility condition between the initial state and the feedback controller at t=0, is achieved by choosing a time varying control operator in a neighbourhood of t=0.
Keywords:Dirichlet control  Navier–  Stokes equations  Feedback control  Stabilization  Riccati equation
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