Spectral analysis of the Schrödinger operator on binary tree-shaped networks and applications

Ammari, Kaïs; Mercier, Denis; Régnier, Virginie

doi:10.1016/j.jde.2015.08.017

Cited by 16 publications

(3 citation statements)

References 23 publications

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“…We have proved that the energy function of the closed-loop system (1) does not decrease exponentially to zero in the case where at least one damping term is absent at the boundary. We expect that this energy may decrease exponentially to a nonvanishing value depending on the initial data (see for example Ammari et al, [9] for the dissipative Schrödinger equation on a binary tree-shaped graph). This will be a good question to address even in the case where we have fewer dampers on the boundary.…”

Section: Riesz Basis Generationmentioning

confidence: 99%

Boundary stabilization for a star-shaped network of variable coefficients strings linked by a point mass

Boughamda¹

2022

DCDS-S

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<p style='text-indent:20px;'>This study is concerned with the pointwise stabilization for a star-shaped network of <inline-formula><tex-math id="M1">\begin{document}$ N $\end{document}</tex-math></inline-formula> variable coefficients strings connected at the common node by a point mass and subject to boundary feedback dampings at all extreme nodes. It is shown that the closed-loop system has a sequence of generalized eigenfunctions which forms a Riesz basis for the state Hilbert space. As a consequence, the spectrum-determined growth condition fulfills. In the meanwhile, the asymptotic expression of the spectrum is presented, and the exponential stability of the system is obtained by giving the optimal decay rate. We prove also that a phenomenon of lack of uniform stability occurs in the absence of damper at one extreme node. This paper reconfirmed the main stability results given by Hansen and Zuazua [SIAM J. Control Optim., <b>33</b> (1995), 1357-1391] in a very particular case.</p>

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Section: Riesz Basis Generationmentioning

confidence: 99%

Boundary stabilization for a star-shaped network of variable coefficients strings linked by a point mass

Boughamda¹

2022

DCDS-S

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“…In the last few years various physical models of multi-link flexible structures consisting of finitely or infinitely many interconnected flexible elements such as strings, beams, plates, shells have been of great interest. See the references [2] as well as [7,4,6] and the references therein. The spectral analysis of such structures has some applications to control or stabilization problems (cf.…”

Section: Introductionmentioning

confidence: 99%

Spectral analysis and stabilization of the dissipative Schrödinger operator on the tadpole graph

Ammari¹,

Assel²

2021

Preprint

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We consider the damped Schrödinger semigroup e −it d 2 dx 2 on the tadpole graph R. We first give a careful spectral analysis and an appropriate decomposition of the kernel of the resolvent. As a consequence and by showing that the generalized eigenfunctions form a Riesz basis of some subspace of L 2 (R), we prove that the corresponding energy decay exponentially. Contents 1. Introduction 2. The Resolvent and the spectrum 2.1. The resolvent 2.2. The spectrum 2.3. Asymptotic of the point spectrum 3. Riesz basis 4. Energy decreasing 4.1. Energy decreasing using the Riesz basis 4.2. Energy decreasing: a numerical example References

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“…Recently Ammari in [3] studied a chain of serially connected strings using a frequency domain method and a special analysis for the resolvent. In [4] the authors analysed the spectrum of the dissipative Schrodinger operator on binary tree-shaped networks, and proved the Riesz basis property of the system. Hence the system satisfies the spectrum determined growth assumption.…”

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confidence: 99%

The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls

Xie¹,

Xu²

2016

NHM

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In this paper, we study the exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls. For the networks, there are some results on the exponential stability, but no result on estimate of the decay rate. The present work mainly estimates the decay rate for these systems, including signal wave equation, serially connected wave equations, and generic tree of 1-d wave equations. By defining the weighted energy functional of the system, and choosing suitable weighted functions, we obtain the estimation value of decay rate of the systems. s(A) = sup{ λ, λ ∈ σ(A)}.

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Spectral analysis of the Schrödinger operator on binary tree-shaped networks and applications

Abstract: In this paper we analyse the spectrum of the dissipative Schrödinger operator on binary tree-shaped networks. As applications, we study the stability of the Schrödinger system using a Riesz basis as well as the transfer function associated to the system.

Cited by 16 publications

References 23 publications

Boundary stabilization for a star-shaped network of variable coefficients strings linked by a point mass

Boundary stabilization for a star-shaped network of variable coefficients strings linked by a point mass

Spectral analysis and stabilization of the dissipative Schrödinger operator on the tadpole graph

The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls

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