## Journals

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### Open Access Journals

NHM

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.

CPAA

We consider $N$ Euler-Bernoulli beams and $N$ strings alternatively connected to one another and forming a particular
network which is a chain beginning with a string.
We study two stabilization problems on the same network and the spectrum of the corresponding conservative system: the characteristic equation as well as its asymptotic behavior are given.
We prove that the energy of the solution of the first dissipative system tends to zero when the time tends to infinity under some irrationality assumptions on the length of the strings and beams.
On another hand we prove a polynomial decay result
of the energy of the second system, independently of the length of the strings and beams, for all regular initial data. Our technique is based on a
frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the
resolvent.

keywords:
Network
,
wave equation
,
spectrum
,
resolvent method
,
Euler-Bernoulli beam equation
,
feedback stabilization.

DCDS

In this paper, we first prove existence
results for general systems of differential equations of parabolic and hyperbolic
type in a Hilbert space setting using the notion of
Agmon-Douglis-Nirenberg elliptic systems on a half-line and finding a necessary
and sufficient condition on the boundary and/or transmission conditions which insures
the dissipativity of the (spatial) operators.
Our second goal is to take advantage of the one-dimensional structure of
networks in order to build appropriate prewavelet bases in view
to the numerical approximation of the above problems. Indeed we
show that the use of such bases for their approximation (by the Galerkin method
for elliptic operators and a fully discrete scheme for parabolic ones)
leads to linear systems which can be preconditioned by a diagonal matrix and
then can be reduced to systems with a condition number uniformly bounded (with
respect to the mesh parameter).

EECT

We consider $N$ strings connected one to another and forming a particular network which is a chain of strings. We study a stabilization problem and precisely we prove that the energy of the solutions of the dissipative system decays exponentially to zero when the time tends to infinity, independently of the densities of the strings. Our technique is based on a
frequency domain method and a special analysis for the resolvent. Moreover,
by the same approach, we study the transfer function associated to the chain of strings and the stability of the Schrödinger system.

keywords:
boundary feedback stabilization.
,
resolvent method
,
transfer function
,
Network
,
wave equation

EECT

We consider a clamped Rayleigh beam equation subject to only one boundary control force. Using an explicit approximation, we first give the asymptotic
expansion of eigenvalues and eigenfunctions of the undamped underlying system. We next establish a polynomial energy decay rate for smooth initial data via an observability inequality of the corresponding undamped problem combined with a boundedness property of the transfer function of the associated undamped problem. Finally, by a frequency domain approach, using the real part of the asymptotic expansion of eigenvalues of the infinitesimal generator of the associated semigroup, we prove that the obtained energy decay rate is optimal.

EECT

In this paper, we consider two damped wave problems for which the damping terms are allowed to change their sign. Using a careful spectral analysis, we find critical values of the damping coefficients for which the problem becomes exponentially or polynomially stable up to these critical values.

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