On the stabilization and controllability for a third order linear  equation

Silva, Patrı́cia Nunes da; Vasconcellos, Carlos Frederico

doi:10.4171/pm/1892

Cited by 4 publications

(5 citation statements)

References 19 publications

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“…Taking into account the work of Silva-Vasconcellos [10], we begin by analysing existence, uniqueness, regularity of solutions and exponential decay of the energy associated to the following system:…”

Section: The Linear Systemmentioning

confidence: 99%

“…We use semigroups theory to prove the existence and uniqueness and to show regularity of solutions we consider the multipliers techniques. The items i), ii), iii) and the dissipation law follow as in Silva-Vasconcellos [10], considering there the parameter γ = 0.…”

Section: The Linear Systemmentioning

confidence: 99%

“…In section 2, for the sake of the completeness, we make a brief analysis of the linear case, that is, the system (1.1) without the term |u| 2 u. We use the work of Silva-Vasconcellos [10] and some multiplier techniques when we consider the additional damping term.…”

Section: Introductionmentioning

confidence: 99%

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The stabilization of the nonlinear model arising in pulse propagation in optical fiber

Silva¹,

Vasconcellos²

2015

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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We analyze the stabilization of the third order nonlinear Schrödinger equation in a bounded interval under the effect of a localized damping mechanism. That is, we consider the following equation: iu t + αu xx + iβu xxx + |u| 2 u + ia(x)u = 0. Where u = u(x, t) is a complex valued function defined in (0, L) × (0, +∞) and α, β are real constants.This equation models a pulse propagation in a long-distance and high-speed optical fiber transmission system. Using multiplier techniques and a special uniform continuation theorem we prove the exponential decay of the total energy associated with the above system.

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Section: The Linear Systemmentioning

confidence: 99%

Section: The Linear Systemmentioning

confidence: 99%

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The stabilization of the nonlinear model arising in pulse propagation in optical fiber

Silva¹,

Vasconcellos²

2015

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

Self Cite

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“…then the L 2 −norm of the solution does not necessarily decay to zero. Here N is the set of critical lengths in the context of exact boundary controllability for the HLS (see [10,14] for the derivation of this set of critical lengths). For instance choosing the coefficients β = 1, α = 2 and δ = 8 with k = 1 and l = 2, we obtain L = π ∈ N .…”

Section: Introductionmentioning

confidence: 99%

Stabilization of higher order Schrödinger equations on a finite interval: Part II

Özsarı

Yılmaz

2022

EECT

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<p style='text-indent:20px;'>Backstepping based controller and observer models were designed for higher order linear and nonlinear Schrödinger equations on a finite interval in [<xref ref-type="bibr" rid="b3">3</xref>] where the controller was assumed to be acting from the left endpoint of the medium. In this companion paper, we further the analysis by considering boundary controller(s) acting at the right endpoint of the domain. It turns out that the problem is more challenging in this scenario as the associated boundary value problem for the backstepping kernel becomes overdetermined and lacks a smooth solution. The latter is essential to switch back and forth between the original plant and the so called target system. To overcome this difficulty we rely on the strategy of using an imperfect kernel, namely one of the boundary conditions in kernel PDE model is disregarded. The drawback is that one loses rapid stabilization in comparison with the left endpoint controllability. Nevertheless, the exponential decay of the <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm with a certain rate still holds. The observer design is associated with new challenges from the point of view of wellposedness and one has to prove smoothing properties for an associated initial boundary value problem with inhomogeneous boundary data. This problem is solved by using Laplace transform in time. However, the Bromwich integral that inverts the transformed solution is associated with certain analyticity issues which are treated through a subtle analysis. Numerical algorithms and simulations verifying the theoretical results are given.</p>

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“…The price to be paid is the lack of any Kato smoothing effect (the system being conservative), which makes the extension of the control results to the nonlinear Boussinesq system more delicate than for KdV. We refer the reader to [7,8,9,10,11,12,13,16,18,19,20,21,27,29,30,31,32,33] for the control and stabilization of KdV, and [14,15,17] for the critical lengths concerning some other dispersive equations.…”

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confidence: 99%

Control of a Boussinesq system of KdV–KdV type on a bounded interval

2019

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We consider a Boussinesq system of KdV-KdV type introduced by J. Bona, M. Chen and J.-C. Saut as a model for the motion of small amplitude long waves on the surface of an ideal fluid. This system of two equations can describe the propagation of waves in both directions, while the single KdV equation is limited to unidirectional waves. We are concerned here with the exact controllability of the Boussinesq system by using some boundary controls. By reducing the controllability problem to a spectral problem which is solved by using the Paley-Wiener method introduced by the third author for KdV, we determine explicitly all the critical lengths for which the exact controllability fails for the linearized system, and give a complete picture of the controllability results with one or two boundary controls of Dirichlet or Neumann type. The extension of the exact controllability to the full Boussinesq system is derived in the energy space in the case of a control of Neumann type. It is obtained by incorporating a boundary feedback in the control in order to ensure a global Kato smoothing effect.2000 Mathematics Subject Classification. Primary: 35Q53, Secondary: 37K10, 93B05, 93D15.

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On the stabilization and controllability for a third order linear equation

Cited by 4 publications

References 19 publications

The stabilization of the nonlinear model arising in pulse propagation in optical fiber

The stabilization of the nonlinear model arising in pulse propagation in optical fiber

Stabilization of higher order Schrödinger equations on a finite interval: Part II

Control of a Boussinesq system of KdV–KdV type on a bounded interval

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