Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings

Kaïs Ammari; Denis Mercier; Virginie Régnier; Julie Valein

doi:10.3934/cpaa.2012.11.785

Article Contents

2012, Volume 11, Issue 2: 785-807. Doi: 10.3934/cpaa.2012.11.785

This issue Previous Article Global solutions to the incompressible magnetohydrodynamic equations Next Article On the structure of the global attractor for non-autonomous dynamical systems with weak convergence

Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings

1.
Département de Mathématiques, Faculté des Sciences de Monastir, 5019 Monastir
2.
LAMAV, FR CNRS 2956, Université de Valenciennes et du Hainaut-Cambrésis, Le Mont Houy, 59313 VALENCIENNES Cedex 9
3.
Univ Lille Nord de France, F-59000 Lille, France, UVHC, LAMAV, FR CNRS 2956, F-59313 Valenciennes
4.
Institut Elie Cartan de Nancy, Université Henri Poincaré, B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cedex

Received: February 28, 2010

Revised: April 30, 2011

Published: February 2012

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Abstract

We consider $N$ Euler-Bernoulli beams and $N$ strings alternatively connected to one another and forming a particular network which is a chain beginning with a string. We study two stabilization problems on the same network and the spectrum of the corresponding conservative system: the characteristic equation as well as its asymptotic behavior are given. We prove that the energy of the solution of the first dissipative system tends to zero when the time tends to infinity under some irrationality assumptions on the length of the strings and beams. On another hand we prove a polynomial decay result of the energy of the second system, independently of the length of the strings and beams, for all regular initial data. Our technique is based on a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.

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Mathematics Subject Classification: Primary: 35L05, 35M10, 35R02; Secondary: 47A10, 93D15, 93D20.

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