Pointwise control of the linearized Gear-Grimshaw system

Capistrano–Filho, Roberto de A.; Komornik, Vilmos; Pazoto, Ademir F.

doi:10.3934/eect.2020029

Cited by 4 publications

(10 citation statements)

References 20 publications

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“…The main difference of the problem we consider in this paper with respect to the above cited papers, except [10], is the coupling. Indeed, in the papers cited above (and in the literature, as far as we know) the coupling is constituted by a zero, first or second order term.…”

Section: Historical Backgroundmentioning

confidence: 99%

“…To conclude, we can mention the work [10] where the authors consider the problem of controlling pointwise, by means of a time dependent Dirac measure supported at a given point, the linear system associated with (1.1) on the unit circle. The results are obtained by means of spectral analysis and Fourier expansion of the solutions.…”

Section: Setting Of the Problemmentioning

confidence: 99%

“…It has also been studied, for example, the controllability in cascade-like systems [29,15], the null controllability in the context of linear thermoelasticity [31], the controllability to trajectories in phase-field models [3], the existence of insensitizing controls for the heat equation [20], and the controllability in reaction-diffusion systems [2]. However, as for KdV-like systems, it is a very recent research topic which has been studied only in [10] and [16].…”

Section: Historical Backgroundmentioning

confidence: 99%

“…In [10], the authors consider the problem of controlling pointwise, by means of a time dependent Dirac measure supported by a given point, the linearized system on the unit circle.…”

Section: Historical Backgroundmentioning

confidence: 99%

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Local null controllability of a model system for strong interaction between internal solitary waves

Bárcena-Petisco

Guerrero

Pazoto

2021

Commun. Contemp. Math.

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In this paper, we prove the local null controllability property for a nonlinear coupled system of two Korteweg–de Vries equations posed on a bounded interval and with a source term decaying exponentially on [Formula: see text]. The system was introduced by Gear and Grimshaw to model the interactions of two-dimensional, long, internal gravity waves propagation in a stratified fluid. We address the controllability problem by means of a control supported on an interior open subset of the domain and acting on one equation only. The proof consists mainly on proving the controllability of the linearized system, which is done by getting a Carleman estimate for the adjoint system. While doing the Carleman, we improve the techniques for dealing with the fact that the solutions of dispersive and parabolic equations with a source term in [Formula: see text] have a limited regularity. A local inversion theorem is applied to get the result for the nonlinear system.

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Section: Historical Backgroundmentioning

confidence: 99%

Section: Setting Of the Problemmentioning

confidence: 99%

Section: Historical Backgroundmentioning

confidence: 99%

“…In [10], the authors consider the problem of controlling pointwise, by means of a time dependent Dirac measure supported by a given point, the linearized system on the unit circle.…”

Section: Historical Backgroundmentioning

confidence: 99%

See 2 more Smart Citations

Local null controllability of a model system for strong interaction between internal solitary waves

Bárcena-Petisco

Guerrero

Pazoto

2021

Commun. Contemp. Math.

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“…The inequality (1.4) is called observability inequality with one moving point in [15]. We refer to [7,14,16,17,23] for more observability from moving points, and to [3,8,21,22] for observability from moving observation domain in higher dimensions.…”

mentioning

confidence: 99%

Observability of dispersive equations from line segments based on graph theory

Yunlei¹,

Wang²

2022

Preprint

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We present a unified treatment on the observability of a family of linear dispersive equations on the torus T. We take one line segment or two line segments in space-time region as the observable set. We give the characteristic on the slopes of the line segments to guarantee the qualitative observability and quantitative observability respectively. The one line segment case is simple, which follows directly from the Ingham's inequality. However, the two line segments case is difficult, the statement of results and the proof rely heavily on the language of graph theory. We also apply our results to (higher order) Schrödinger equations and the linear KdV equation.

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Global control aspects for long waves in nonlinear dispersive media

Capistrano–Filho

Gomes²

2023

ESAIM: COCV

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A class of models of long waves in dispersive media with coupled quadratic nonlinearities on a periodic domain $\mathbb{T}$ are studied. We used two distributed controls, supported in $\omega\subset\mathbb{T}$ and assumed to be generated by a linear feedback law conserving the \textit { ``mass" (or ``volume")}, to prove global control results. The first result, using spectral analysis, guarantees that the system in consideration is locally controllable in $H^s(\mathbb{T})$ , for $s\geq0$ . After that, by certain properties of Bourgain spaces, we show a property of global exponential stability. This property together with the local exact controllability ensures for the first time in the literature that long waves in nonlinear dispersive media are globally exactly controllable in large time. Precisely, our analysis relies strongly on the \textit { bilinear estimates} using the Fourier restriction spaces in two different dispersions that will guarantee a global control result for coupled systems of the Korteweg–de Vries type. This result, of independent interest in the area of control of coupled dispersive systems, provides a necessary first step for the study of global control properties to the coupled dispersive systems in periodic domains.

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Pointwise control of the linearized Gear-Grimshaw system

Cited by 4 publications

References 20 publications

Local null controllability of a model system for strong interaction between internal solitary waves

Local null controllability of a model system for strong interaction between internal solitary waves

Observability of dispersive equations from line segments based on graph theory

Global control aspects for long waves in nonlinear dispersive media

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