Feedback stabilization of a coupled string-beam system

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March 2009, 4(1): 19-34. doi: 10.3934/nhm.2009.4.19

Feedback stabilization of a coupled string-beam system

Kaïs Ammari ^1, , Mohamed Jellouli ^1, and Michel Mehrenberger ^2,

1.	Département de Mathématiques, Faculté des Sciences de Monastir, 5019 Monastir, Tunisia, Tunisia
2.	Institut de Recherche Mathématique Avancée, Université Louis Pasteur, 7, rue René Descartes, 67084 Strasbourg, France

Received December 2007 Revised September 2008 Published February 2009

Abstract
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We consider a stabilization problem for a coupled string-beam system. We prove some decay results of the energy of the system. The method used is based on the methodology introduced in Ammari and Tucsnak [2] where the exponential and weak stability for the closed loop problem is reduced to a boundedness property of the transfer function of the associated open loop system. Morever, we prove, for the same model but with two control functions, independently of the length of the beam that the energy decay with a polynomial rate for all regular initial data. The method used, in this case, is based on a frequency domain method and combine a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.

Keywords: Feedback stabilization, Observability, Coupled string-beam system..

Mathematics Subject Classification: Primary: 93B07, 35M10, 35A25; Secondary: 42A1.

Citation: Kaïs Ammari, Mohamed Jellouli, Michel Mehrenberger. Feedback stabilization of a coupled string-beam system. Networks & Heterogeneous Media, 2009, 4 (1) : 19-34. doi: 10.3934/nhm.2009.4.19

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