Spectrum analysis of a serially connected Euler-Bernoulli beams problem

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.