Global stabilization of the Navier-Stokes equations around an unstable equilibrium statewith a boundary feedback controller

Evrad M. D. Ngom; Abdou Sène; Daniel Y. Le Roux

doi:10.3934/eect.2015.4.89

Article Contents

2015, Volume 4, Issue 1: 89-106. Doi: 10.3934/eect.2015.4.89

This issue Previous Article The $L^p$-approach to the fluid-rigid body interaction problem for compressible fluids Next Article Backward uniqueness for linearized compressible flow

Global stabilization of the Navier-Stokes equations around an unstable equilibrium state with a boundary feedback controller

1.
UFR de Sciences Appliquées et Technologie, Université Gaston Berger, B.P. 234 Saint-Louis
2.
CEA-MITIC, Université Gaston Berger, B.P. 234 Saint-Louis
3.
Université de Lyon, CNRS, Université Lyon 1, Institut Camille Jordan, 43, blvd du 11 novembre 1918, 69622 Villeurbanne Cedex

Received: July 31, 2014

Revised: November 30, 2014

Published: February 2015

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Abstract

This paper presents a global stabilization for the two and three-dimensional Navier-Stokes equations in a bounded domain $\Omega$ around a given unstable equilibrium state, by means of a boundary normal feedback control. The control is expressed in terms of the velocity field by using a non-linear feedback law. In order to determine the feedback control law, we consider an extended system coupling the equations governing the perturbation with an equation satisfied by the control on the domain boundary. By using the Faedo-Galerkin method and a priori estimation techniques, a stabilizing boundary control is built. This control law ensures a decrease of the energy of the controlled discrete system. A compactness result then allows us to pass to the limit in the system satisfied by the approximated solutions.

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Mathematics Subject Classification: Primary: 37L65, 49J99, 76D55; Secondary: 76D05.

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