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Evolution Equations and Control Theory (EECT)
 

Boundary stabilization of the Navier-Stokes equations with feedback controller via a Galerkin method

Pages: 147 - 166, Volume 3, Issue 1, March 2014      doi:10.3934/eect.2014.3.147

 
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Evrad M. D. Ngom - UFR de Sciences Appliquées et Technologie, Université Gaston Berger, B.P. 234 Saint-Louis, Senegal (email)
Abdou Sène - UFR de Sciences Appliquées et Technologie, Université Gaston Berger, B.P. 234 Saint-Louis, Senegal (email)
Daniel Y. Le Roux - Université de Lyon, CNRS, Université Lyon 1, Institut Camille Jordan, 43, blvd du 11 novembre 1918, 69622 Villeurbanne Cedex, France (email)

Abstract: In this work we study the exponential stabilization of the two and three-dimensional Navier-Stokes equations in a bounded domain $\Omega$, around a given steady-state flow, by means of a boundary control. In order to determine a feedback law, we consider an extended system coupling the Navier-Stokes equations with an equation satisfied by the control on the domain boundary. While most traditional approaches apply a feedback controller via an algebraic Riccati equation, the Stokes-Oseen operator or extension operators, a Galerkin method is proposed instead in this study. The Galerkin method permits to construct a stabilizing boundary control and by using energy a priori estimation technics, the exponential decay is obtained. A compactness result then allows us to pass to the limit in the system satisfied by the approximated solutions. The resulting feedback control is proven to be globally exponentially stabilizing the steady states of the two and three-dimensional Navier-Stokes equations.

Keywords:  Navier-Stokes system, feedback control, boundary stabilization, Galerkin method.
Mathematics Subject Classification:  Primary: 37L65, 49J99, 76D55; Secondary: 76D05.

Received: June 2013;      Revised: November 2013;      Available Online: February 2014.

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