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Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system

Published online by Cambridge University Press:  21 October 2008

Mehdi Badra
Mehdi Badra*
Affiliation:
Laboratoire LMA, UMR CNRS 5142, Université de Pau et des Pays de l'Adour, 64013 Pau Cedex, France. mehdi.badra@univ-pau.fr
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Abstract

We study the local exponential stabilization of the 2D and 3DNavier-Stokes equations in a bounded domain, around a givensteady-state flow, by means of a boundary control. We look for acontrol so that the solution to the Navier-Stokes equations be astrong solution. In the 3D case, such solutions may exist if theDirichlet control satisfies a compatibility condition with theinitial condition. In order to determine a feedback law satisfyingsuch a compatibility condition, we consider an extended systemcoupling the Navier-Stokes equations with an equation satisfied bythe control on the boundary of the domain. We determine a linearfeedback law by solving a linear quadratic control problem for thelinearized extended system. We show that this feedback law alsostabilizes the nonlinear extended system.

Keywords

Navier-Stokes equationfeedback stabilizationDirichlet controlRiccati equation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 4 , October 2009 , pp. 934 - 968
Copyright
© EDP Sciences, SMAI, 2008

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