Spectrum analysis of a serially connected Euler-Bernoulli beams problem
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LAMAV, FR CNRS 2956, Université de Valenciennes et du Hainaut-Cambrésis, Le Mont Houy, 59313 VALENCIENNES Cedex 9
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Received:
01 April 2009
Revised:
01 August 2009
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34B45, 74K10, 93B60, 93B05.
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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.
Citation: Denis Mercier. Spectrum analysis of a serially connected Euler-Bernoulli beams problem[J]. Networks and Heterogeneous Media, 2009, 4(4): 709-730. doi: 10.3934/nhm.2009.4.709
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Abstract
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.
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