Uniform stabilization of a nonlinear dispersive system

Pazoto, Ademir F.; Souza, G. R.

doi:10.1090/s0033-569x-2013-01343-1

Cited by 7 publications

(10 citation statements)

References 25 publications

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“…It has the structure of a pair of KdV equations with both linear and nonlinear coupling terms and has been object of intensive research in recent years. In what concerns the stabilization problems, most of the works have been focused on a bounded interval with a localized internal damping (see, for instance, [14] and the references therein). In particular, we also refer to [1] for an extensive discussion on the physical relevance of the system and to [3,4,5,6,7] for the results used in this paper.…”

Section: Introductionmentioning

confidence: 99%

Stabilization of the Gear–Grimshaw system on a periodic domain

Capistrano–Filho

Komornik

Pazoto

2014

Commun. Contemp. Math.

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Abstract. This paper is devoted to the study of a nonlinear coupled system of two Korteweg-de Vries equations in a periodic domain under the effect of an internal damping term. The system was introduced Gear and Grimshaw to model the interactions of two-dimensional, long, internal gravity waves propagation in a stratified fluid. Designing a time-varying feedback law and using a Lyapunov approach we establish the exponential stability of the solutions in Sobolev spaces of any positive integral order.

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Section: Introductionmentioning

confidence: 99%

Stabilization of the Gear–Grimshaw system on a periodic domain

Capistrano–Filho

Komornik

Pazoto

2014

Commun. Contemp. Math.

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“…In what concerns the stabilization problems, most of the works have been focused on a bounded interval with a localized internal damping (see, for instance, [12] and the references therein). However, the stabilization results for system (1.1)-(1.2) was first obtained in [4], when the authors considered the system in a periodic domain.…”

Section: State Of Artmentioning

confidence: 99%

Stabilization of the Gear–Grimshaw System with Weak Damping

Capistrano–Filho

2017

J Dyn Control Syst

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Abstract. The aim of this work is to consider the internal stabilization of a nonlinear coupled system of two Korteweg-de Vries equations in a finite interval under the effect of a very weak localized damping. The system was introduced by Gear and Grimshaw to model the interactions of two-dimensional, long, internal gravity waves propagation in a stratified fluid. Considering feedback controls laws and using "Compactness-Uniqueness Argument", which reduce the problem to use a unique continuation property, we establish the exponential stability of the weak solutions when the exponent in the nonlinear term ranges over the interval [1,4).

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“…The internal stabilization problem has also been addressed (see, for instance, [4,9,29] and the references therein). Although controllability and stabilization problems are closely related, one may expect that some of the available results will have some counterparts in the context of the control problem, but this issue is open.…”

Section: Introductionmentioning

confidence: 99%

Pointwise control of the linearized Gear-Grimshaw system

Capistrano–Filho¹,

Komornik²,

Pazoto³

2020

Evolution Equations &Amp; Control Theory

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In this paper we consider the problem of controlling pointwise, by means of a time dependent Dirac measure supported by a given point, a coupled system of two Korteweg-de Vries equations on the unit circle. More precisely, by means of spectral analysis and Fourier expansion we prove, under general assumptions on the physical parameters of the system, a pointwise observability inequality which leads to the pointwise controllability when we observe two control functions. In addition, with a uniqueness property proved for the linearized system without control, we are also able to show pointwise controllability when only one control function acts internally. In both cases we can find, under some assumptions on the coefficients of the system, the sharp time of the controllability.Date: Version 2019-06-16. 2010 Mathematics Subject Classification. Primary: 93B07, 35Q53 Secondary: 93B52, 93B05.

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Uniform stabilization of a nonlinear dispersive system

Cited by 7 publications

References 25 publications

Stabilization of the Gear–Grimshaw system on a periodic domain

Stabilization of the Gear–Grimshaw system on a periodic domain

Stabilization of the Gear–Grimshaw System with Weak Damping

Pointwise control of the linearized Gear-Grimshaw system

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