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December  2009, 4(4): 709-730. doi: 10.3934/nhm.2009.4.709

Spectrum analysis of a serially connected Euler-Bernoulli beams problem

Denis Mercier 1,

1. 

LAMAV, FR CNRS 2956, Université de Valenciennes et du Hainaut-Cambrésis, Le Mont Houy, 59313 VALENCIENNES Cedex 9, France

Received  April 2009 Revised  August 2009 Published  October 2009

In this article we analyse the eigenfrequencies of a hyperbolic system which corresponds to a chain of Euler-Bernoulli beams. More precisely we show that the distance between two consecutive large eigenvalues of the spatial operator involved in this evolution problem is superior to a minimal fixed value. This property called spectral gap holds as soon as the roots of a function denoted by $f_{\infty}$ (and giving the asymptotic behaviour of the eigenvalues) are all simple. For a chain of $N$ different beams, this assumption on the multiplicity of the roots of $f_{\infty}$ is proved to be satisfied. A direct consequence of this result is that we obtain the exact controllability of an associated boundary controllability problem. It is well-known that the spectral gap is a important key point in order to get the exact controllabilty of these one-dimensional problem and we think that the new method developed in this paper could be applied in other related problems.
Keywords: spectral gap, beams, Network, controllability., eigenvalue.
Mathematics Subject Classification: 34B45, 74K10, 93B60, 93B0.
Citation: Denis Mercier. Spectrum analysis of a serially connected Euler-Bernoulli beams problem. Networks & Heterogeneous Media, 2009, 4 (4) : 709-730. doi: 10.3934/nhm.2009.4.709
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