Stabilization of the Schrödinger equation with a delay term in boundary feedback or internal feedback

Nicaise, Serge; Rebiai, Salah-Eddine

doi:10.4171/pm/1879

Cited by 23 publications

(8 citation statements)

References 13 publications

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“…Nicaise and Pignotti [6] extended this result to the multi-dimensional wave equation with a delay term in the boundary or internal feedbacks. The same type of result was obtained by Nicaise and Rebiai [7] for the Schrödinger equation.…”

Section: Introductionsupporting

confidence: 83%

Exponential Stability of Compactly Coupled Wave Equations with Delay Terms in the Boundary Feedbacks

Rebiai

Ali

2014

IFIP Advances in Information and Communication Technology

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International audienceWe consider a linear system of compactly coupled wave equations with Neumann feedback controllers that contain delay terms. First, we prove under some assumptions that the closed-loop system generates a $$C_{0}-$$semigroup of contractions on an appropriate Hilbert space. Then, under further assumptions, we show that the closed-loop system is exponentially stable. This result is obtained by introducing a suitable energy function and by using an observability estimate

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Section: Introductionsupporting

confidence: 83%

Exponential Stability of Compactly Coupled Wave Equations with Delay Terms in the Boundary Feedbacks

Rebiai

Ali

2014

IFIP Advances in Information and Communication Technology

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“…the fact that the weight of the term with delay is smaller than the weight of the term without delay, can be found in [16], for the asymptotic stability of the wave equation with delayed feedback. This restriction about the weights of the feedbacks is also used for hyperbolic and parabolic partial differential equations in [19,20] and even for the Schrödinger equation (which is a dispersive equation, like KdV equation) in [18]. This restrictive assumption is necessary in these cases and if they are not satisfied, it can be shown that instabilities may appear (see for instance [11], [12] with a = 0, or [16] in the more general case for the wave equation).…”

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confidence: 99%

On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback

Valein

2022

MCRF

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<p style='text-indent:20px;'>The aim of this work is to study the asymptotic stability of the nonlinear Korteweg-de Vries equation in the presence of a delayed term in the internal feedback. We first consider the case where the weight of the term with delay is smaller than the weight of the term without delay and we prove a semiglobal stability result for any lengths. Secondly we study the case where the support of the term without delay is not included in the support of the term with delay. In that case, we give a local exponential stability result if the weight of the delayed term is small enough. We illustrate these results by some numerical simulations.</p>

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“…Note also that, in [NP06], dealing with the wave equation, there are some restrictions about the positive coefficients of the terms with or without delay. Actually, it is the case for hyperbolic and parabolic partial differential equations in [NV10], [NVF09] and even for the Schrödinger equation (which is a dispersive equation, just like KdV) in [NR11]. In these papers, the authors assume that the coefficient of the term with delay is smaller than the coefficient of the term without delay, i.e.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Two Approaches for the Stabilization of Nonlinear KdV Equation With Boundary Time-Delay Feedback

Baudouin

Crépeau

Valein

2019

IEEE Trans. Automat. Contr.

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This article concerns the nonlinear Korteweg-de Vries equation with boundary timedelay feedback. Under appropriate assumption on the coefficients of the feedbacks (delayed or not), we first prove that this nonlinear infinite dimensional system is well-posed for small initial data. The main results of our study are two theorems stating the exponential stability of the nonlinear time delay system. Two different methods are employed: a Lyapunov functional approach (allowing to have an estimation on the decay rate, but with a restrictive assumption on the length of the spatial domain of the KdV equation) and an observability inequality approach, with a contradiction argument (for any non critical lengths but without estimation on the decay rate). Some numerical simulations are given to illustrate the results.

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Stabilization of the Schrödinger equation with a delay term in boundary feedback or internal feedback

Cited by 23 publications

References 13 publications

Exponential Stability of Compactly Coupled Wave Equations with Delay Terms in the Boundary Feedbacks

Exponential Stability of Compactly Coupled Wave Equations with Delay Terms in the Boundary Feedbacks

On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback

Two Approaches for the Stabilization of Nonlinear KdV Equation With Boundary Time-Delay Feedback

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