Exact Controllability of Serially Connected Euler Bernoulli Beams

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Exact Controllability of Serially Connected Euler-Bernoulli Beams Denis Mercier, ? March 25, 2008 Abstract We consider the exact controllability problem by boundary action of hyperbolic sys- tems of a chain of Euler-Bernoulli beams. By the classical Hilbert Uniqueness Method, the control problem is reduced to the obtention of an observability inequality. So that we need to study the asymptotic behaviour of the eigenvalues. 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∞ (and giving the asymptotic behaviour of the eigenvalues) are all simple. For a chain of N different beams, this as- sumption on the multiplicity of the roots of f∞ is proved to be satisfied. From this result and some estimate concerning the eigenvectors controllability follows. Key words Network, Beams, Eigenvalue, Spectral Gap, Controllability. AMS 34B45, 74K10, 93B60, 93B05. 1 Introduction In the last few years various physical models of multi-link flexible structures consisting of finitely many interconnected flexible elements such as strings, beams, plates, shells have been mathematically studied. See [7], [8], [13], [20], [22] for instance.

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Exact Controllability of Serially Connected Euler-Bernoulli Beams

Denis Mercier, ∗ March 25, 2008

Abstract We consider the exact controllability problem by boundary action of hyperbolic sys-tems of a chain of Euler-Bernoulli beams. By the classical Hilbert Uniqueness Method, the control problem is reduced to the obtention of an observability inequality. So that we need to study the asymptotic behaviour of the eigenvalues. 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 ∞ (and giving the asymptotic behaviour of the eigenvalues) are all simple. For a chain of N diﬀerent beams, this as-sumption on the multiplicity of the roots of f ∞ is proved to be satisﬁed. From this result and some estimate concerning the eigenvectors controllability follows. Key words Network, Beams, Eigenvalue, Spectral Gap, Controllability. AMS 34B45, 74K10, 93B60, 93B05.

1 Introduction In the last few years various physical models of multi-link ﬂexible structures consisting of ﬁnitely many interconnected ﬂexible elements such as strings, beams, plates, shells have been mathematically studied. See [7], [8], [13], [20], [22] for instance. The spectral analysis of such structures has some applications to control or stabilization problems ([20] and [21]). For inter-connected strings (corresponding to a second-order operator on each string), a lot of results have been obtained: the asymptotic behaviour of the eigenvalues ([1], [2], [6], [28]), the relationship between the eigenvalues and algebraic theory (cf. [3], [4], [20], [27]), qualitative properties of solutions (see [6] and [30]) and ﬁnally studies of the Green function (cf. [18], [31], [32]). For interconnected beams (corresponding to a fourth-order operator on each beam), some results on the asymptotic behaviour of the eigenvalues and on the relationship between the eigenvalues ´ ∗ Laboratoire de Mathematiques et ses Applications de Valenciennes, FR CNRS 2956, Institut des Sciences etTechniquesdeValenciennes,Universit´edeValenciennesetduHainaut-Cambre´sis,LeMontHouy,59313 VALENCIENNES Cedex 9, FRANCE, e-mail : denis.mercier@univ-valenciennes.fr

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