Stabilization and Control for the Biharmonic Schrödinger Equation

Capistrano–Filho, Roberto de A.; Cavalcante, Márcio

doi:10.1007/s00245-019-09640-8

Cited by 17 publications

(12 citation statements)

References 35 publications

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“…More precisely, they proved that on torus T, the solution of the associated linear system (1) is globally exponential stable, by using certain properties of propagation of compactness and regularity in Bourgain spaces. This property, together with the local exact controllability, ensures that fourth order nonlinear Schrödinger is globally exactly controllable, we suggest the reader to see [15] for more details.…”

Section: Introductionmentioning

confidence: 91%

“…Recently, in [15], the first and the second authors worked with equation ( 1) with the purpose to obtain controllability results. More precisely, they proved that on torus T, the solution of the associated linear system (1) is globally exponential stable, by using certain properties of propagation of compactness and regularity in Bourgain spaces.…”

Section: Introductionmentioning

confidence: 99%

“…We are now generalize the boundary forcing operator (15). For Re λ > −4 and given g ∈ C ∞ 0 (R + ), we define…”

mentioning

confidence: 99%

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Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation

Capistrano–Filho

Cavalcante

Gallego

2022

DCDS-B

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In a recent article [<xref ref-type="bibr" rid="b16">16</xref>], the authors gave a starting point of the study on a series of problems concerning the initial boundary value problem and control theory of Biharmonic NLS in some non-standard domains. In this direction, this article deals to present answers for some questions left in [<xref ref-type="bibr" rid="b16">16</xref>] concerning the study of the cubic fourth order Schrödinger equation in a star graph structure <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{G} $\end{document}</tex-math></inline-formula>. Precisely, consider <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{G} $\end{document}</tex-math></inline-formula> composed by <inline-formula><tex-math id="M3">\begin{document}$ N $\end{document}</tex-math></inline-formula> edges parameterized by half-lines <inline-formula><tex-math id="M4">\begin{document}$ (0,+\infty) $\end{document}</tex-math></inline-formula> attached with a common vertex <inline-formula><tex-math id="M5">\begin{document}$ \nu $\end{document}</tex-math></inline-formula>. With this structure the manuscript proposes to study the well-posedness of a dispersive model on star graphs with three appropriated vertex conditions by using the boundary forcing operator approach. More precisely, we give positive answer for the Cauchy problem in low regularity Sobolev spaces. We have noted that this approach seems very efficient, since this allows to use the tools of Harmonic Analysis, for instance, the Fourier restriction method, introduced by Bourgain, while for the other known standard methods to solve partial differential partial equations on star graphs are more complicated to capture the dispersive smoothing effect in low regularity. The arguments presented in this work have prospects to be applied for other nonlinear dispersive equations in the context of star graphs with unbounded edges.

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Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

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Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation

Capistrano–Filho

Cavalcante

Gallego

2022

DCDS-B

Self Cite

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“…Several results concerning well-posedness for (6.5) may be found in [10] (see also subsequent references). Control and stabilization for (6.5) have already appeared in [4]. Equation (6.4) also serves as the linear version of the more general equation…”

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confidence: 99%

Two simple criterion to obtain exact controllability and stabilization of a linear family of dispersive PDE's on a periodic domain

Leal

Pastor

2022

EECT

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In this work, we use the classical moment method to find a practical and simple criterion to determine if a family of linearized Dispersive equations on a periodic domain is exactly controllable and exponentially stabilizable with any given decay rate in <inline-formula><tex-math id="M1">\begin{document}$ H_{p}^{s}(\mathbb{T}) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ s\in \mathbb{R}. $\end{document}</tex-math></inline-formula> We apply these results to prove that the linearized Smith equation, the linearized dispersion-generalized Benjamin-Ono equation, the linearized fourth-order Schrödinger equation, and the Higher-order Schrödinger equations are exactly controllable and exponentially stabilizable with any given decay rate in <inline-formula><tex-math id="M3">\begin{document}$ H_{p}^{s}(\mathbb{T}) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ s\in \mathbb{R}. $\end{document}</tex-math></inline-formula>

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“…During the last 40 years, there have been many contributions to the exact controllability, stabilizability, and the boundary control (linear and nonlinear) for differents models like the KdV, the improved Boussinesq equation, the Boussinesq equation with variable coefficients, the generalized Boussinesq equation, the Schrödinger equation (see for example, D. Russell in [15], B. Zhang in [19,20], D. Russell and B. Zhang in [16,17], D. Rosier in [13], M. Chapouly in [6], E. Cerpa and E. Crépeau in [4], E. Crépeau in [7], E. Cerpa and I. Rivas in [5], J. Amara and H. Bouzidi in [1], R. Capistrano and M. Cavalcante in [3], among others).…”

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confidence: 99%

On the exact controllability and the stabilization for the Benney-Luke equation

Quintero¹,

Montes²

2019

Mathematical Control &Amp; Related Fields

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In this work we consider the exact controllability and the stabilization for the generalized Benney-Luke equationon a periodic domain S (the unit circle on the plane) with internal control f supported on an arbitrary sub-domain of S. We establish that the model is exactly controllable in a Sobolev type space when the whole S is the support of f , without any assumption on the size of the initial and final states, and that the model is local exactly controllable when the support of f is a proper subdomain of S, assuming that initial and terminal states are small. Moreover, in the case that the initial data is small and f is a special internal linear feedback, the solution of the model must have uniform exponential decay to a constant state.

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Stabilization and Control for the Biharmonic Schrödinger Equation

Cited by 17 publications

References 35 publications

Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation

Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation

Two simple criterion to obtain exact controllability and stabilization of a linear family of dispersive PDE's on a periodic domain

On the exact controllability and the stabilization for the Benney-Luke equation

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