Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings
Kaïs Ammari Denis Mercier Virginie Régnier Julie Valein
Communications on Pure & Applied Analysis 2012, 11(2): 785-807 doi: 10.3934/cpaa.2012.11.785
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.
Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks
Serge Nicaise Julie Valein
Networks & Heterogeneous Media 2007, 2(3): 425-479 doi: 10.3934/nhm.2007.2.425
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.
keywords: wave equation stabilization delay
Stability of the heat and of the wave equations with boundary time-varying delays
Serge Nicaise Julie Valein Emilia Fridman
Discrete & Continuous Dynamical Systems - S 2009, 2(3): 559-581 doi: 10.3934/dcdss.2009.2.559
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.
keywords: stability Heat equation wave equation time-varying delay Lyapunov functional.
Exponential stability of the wave equation with boundary time-varying delay
Serge Nicaise Cristina Pignotti Julie Valein
Discrete & Continuous Dynamical Systems - S 2011, 4(3): 693-722 doi: 10.3934/dcdss.2011.4.693
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.
keywords: stabilization. Wave equation delay feedbacks
Global Carleman estimate on a network for the wave equation and application to an inverse problem
Lucie Baudouin Emmanuelle Crépeau Julie Valein
Mathematical Control & Related Fields 2011, 1(3): 307-330 doi: 10.3934/mcrf.2011.1.307
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.
keywords: Networks Carleman estimate. inverse problem wave equation

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