Feedback boundary stabilization of 2d fluid-structure interaction systems

Mehdi Badra; Takéo Takahashi

doi:10.3934/dcds.2017102

Article Contents

2017, Volume 37, Issue 5: 2315-2373. Doi: 10.3934/dcds.2017102

This issue Previous Article Control systems on flag manifolds and their chain control sets Next Article Homogenization of trajectory attractors of 3D Navier-Stokes system with randomly oscillating force

Feedback boundary stabilization of 2d fluid-structure interaction systems

Mehdi Badra^1, , and
Takéo Takahashi^2,

1.
CNRS / Univ Pau & Pays Adour, Laboratoire de Mathématiques et, de leurs Applications de Pau -Fédération IPRA, UMR5142, 64000, Pau, France
2.
Institut Élie Cartan, UMR 7502, INRIA, Nancy-Université, CNRS, BP239, 54506 Vandœuvre-lès-Nancy Cedex, France, Team SPHINX. INRIA Nancy -Grand Est, 615, rue du Jardin Botanique, 54600 Villers-lès-Nancy, France

* Corresponding author: Mehdi Badra
* Corresponding author: Mehdi Badra

Received: April 2016

Revised: December 2016

Published: May 2017

The authors are partially supported by the project ANR IFSMACS (ANR-15-CE40-0010) financed by the French Agence Nationale de la Recherche.

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Abstract

We study the feedback stabilization of a system composed by an incompressible viscous fluid and a deformable structure located at the boundary of the fluid domain. We stabilize the position and the velocity of the structure and the velocity of the fluid around a stationary state by means of a Dirichlet control, localized on the exterior boundary of the fluid domain and with values in a finite dimensional space. Our result concerns weak solutions for initial data close to the stationary state. Our method is based on general arguments for stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domain of the stationary state and of the stabilized solution are different. We prove that for initial data close to the stationary state, we can stabilize the position and the velocity of the deformable structure and the velocity of the fluid.

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Mathematics Subject Classification: Primary: 93C20, 93D15, 74F10; Secondary: 76D55, 76D05, 35Q30.

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Figure 1. The fluid-plate system

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