Spectrum analysis of a serially connected Euler-Bernoulli beams problem
Denis Mercier
Networks & Heterogeneous Media 2009, 4(4): 709-730 doi: 10.3934/nhm.2009.4.709
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.
keywords: spectral gap beams Network controllability. eigenvalue
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.
Existence results for general systems of differential equations on one-dimensional networks and prewavelets approximation
Denis Mercier Serge Nicaise
Discrete & Continuous Dynamical Systems - A 1998, 4(2): 273-300 doi: 10.3934/dcds.1998.4.273
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).
keywords: systems Networks differential equations prewavelets.
Boundary feedback stabilization of a chain of serially connected strings
Kaïs Ammari Denis Mercier
Evolution Equations & Control Theory 2015, 4(1): 1-19 doi: 10.3934/eect.2015.4.1
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
Optimal energy decay rate of Rayleigh beam equation with only one boundary control force
Maya Bassam Denis Mercier Ali Wehbe
Evolution Equations & Control Theory 2015, 4(1): 21-38 doi: 10.3934/eect.2015.4.21
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.
keywords: polynomial stability Rayleigh beam equation boundary stabilization optimal energy decay.
Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems
Farah Abdallah Denis Mercier Serge Nicaise
Evolution Equations & Control Theory 2013, 2(1): 1-33 doi: 10.3934/eect.2013.2.1
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.
keywords: Indefinite damping spectral analysis. Riesz basis

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