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### Open Access Journals

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.

NHM

In this paper we consider the wave equation on 1-d networks with a delay term in the boundary and/or transmission conditions. We first show the well posedness of the problem and the decay of an appropriate energy. We give a necessary and sufficient condition that guarantees the decay to zero of the energy. We further give sufficient conditions that lead to exponential or polynomial stability of the solution. Some examples are also given.

DCDS-S

Exponential stability analysis via
Lyapunov method is extended to the one-dimensional heat and wave equations
with time-varying delay in the boundary conditions.
The delay function is admitted to be time-varying
with an

*a priori*given upper bound on its derivative, which is less than $1$. Sufficient and explicit conditions are derived that guarantee the exponential stability. Moreover the decay rate can be explicitly computed if the data are given.
DCDS-S

We consider the wave equation with a time-varying delay term in
the boundary condition in a bounded and smooth domain $\Omega\subset\RR^n.$ Under suitable
assumptions, we prove exponential stability of the solution.
These results are obtained by introducing suitable energies and suitable Lyapunov functionals. Such analysis is also extended to a nonlinear version
of the model.

MCRF

We are interested in an inverse problem for the wave equation with potential on a star-shaped network. We prove the Lipschitz stability of the inverse problem consisting in the determination of the potential on each string of the network with Neumann boundary measurements at all but one external vertices. Our main tool, proved in this article, is a global Carleman estimate for the network.

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