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

Pages: 785 - 807, Volume 11, Issue 2, March 2012

doi:10.3934/cpaa.2012.11.785       Abstract        References        Full Text (452.1K)       Related Articles

Kaïs Ammari - Département de Mathématiques, Faculté des Sciences de Monastir, 5019 Monastir, Tunisia (email)
Denis Mercier - LAMAV, FR CNRS 2956, Université de Valenciennes et du Hainaut-Cambrésis, Le Mont Houy, 59313 VALENCIENNES Cedex 9, France (email)
Virginie Régnier - Univ Lille Nord de France, F-59000 Lille, France, UVHC, LAMAV, FR CNRS 2956, F-59313 Valenciennes, France (email)
Julie Valein - Institut Elie Cartan de Nancy, Université Henri Poincaré, B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France (email)

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.

Keywords:  Network, wave equation, Euler-Bernoulli beam equation, spectrum, resolvent method, feedback stabilization.
Mathematics Subject Classification:  Primary: 35L05, 35M10, 35R02; Secondary: 47A10, 93D15, 93D20.

Received: March 2010;      Revised: May 2011;      Published: October 2011.

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