Exact boundary controllability of the nonlinear Schrödinger equation

Rosier, Lionel; Zhang, Bingyu

doi:10.1016/j.jde.2008.11.004

Cited by 52 publications

(31 citation statements)

References 27 publications

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“…We should also mention some important work related to the control and stabilization of the KdV equation. Exact boundary controllability of the linear and nonlinear KdV equations with the same type of boundary conditions as in (1) was studied by [22], [7], [27], [9], [3], [5], [23], and [10]. Stabilization of solutions of the KdV equation with a localised interior damping was achieved by [21], [20], [17], and [1].…”

Section: A Few More Words On the Literaturementioning

confidence: 99%

Output feedback stabilization of the linearized Korteweg–de Vries equation with right endpoint controllers

Batal

Özsarı

2019

Automatica

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In this paper, we prove the output feedback stabilization for the linearized Korteweg-de Vries (KdV) equation posed on a finite domain in the case the full state of the system cannot be measured. We assume that there is a sensor at the left end point of the domain capable of measuring the first and second order boundary traces of the solution. This allows us to design a suitable observer system whose states can be used for constructing boundary feedbacks acting at the right endpoint so that both the observer and the original plant become exponentially stable. Stabilization of the original system is proved in the L 2 -sense, while the convergence of the observer system to the original plant is also proved in higher order Sobolev norms. The standard backstepping approach used to construct a left endpoint controller fails and presents mathematical challenges when building right endpoint controllers due to the overdetermined nature of the related kernel models. In order to deal with this difficulty we use the method of [18] which is based on using modified target systems involving extra trace terms. In addition, we show that the number of controllers and boundary measurements can be reduced to one, with the cost of a slightly lower exponential rate of decay. We provide numerical simulations illustrating the efficacy of our controllers. of the following problem (so called "target system"):in Ω × R + , w(0, t) = w(L, t) = w x (L, t) = 0 in R + , w(x, 0) = w 0 (x) ≡ u 0 − x 0 k(x, y)u 0 (y)dy in Ω.

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Section: A Few More Words On the Literaturementioning

confidence: 99%

Output feedback stabilization of the linearized Korteweg–de Vries equation with right endpoint controllers

Batal

Özsarı

2019

Automatica

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“…In the last section, using existing software, we show explicit computations and an explicit representation of the observability constant for a variety of potentials including a damped harmonic oscillator. While the problem for the non-linear Schrödinger equation has been investigated from the control theory standpoint [24,25,36,37], to the best of the authors knowledge, the problem of observability for potentials has not been addressed in an explicit way using eigenfunctions and numerical methods. Observability for the linear Schrödinger equation was examined in [34].…”

Section: One Can Instead Consider a Time Asymptotic Observability Conmentioning

confidence: 99%

Asymptotics for Optimal Design Problems for the Schrödinger Equation with a Potential

Waters

Merkurjev

2018

Journal of Optimization

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We study the problem of optimal observability and prove time asymptotic observability estimates for the Schrödinger equation with a potential in L ∞ (Ω), with Ω ⊂ R d , using spectral theory. An elegant way to model the problem using a time asymptotic observability constant is presented. For certain small potentials, we demonstrate the existence of a nonzero asymptotic observability constant under given conditions and describe its explicit properties and optimal values. Moreover, we give a precise description of numerical models to analyze the properties of important examples of potentials wells, including that of the modified harmonic oscillator.

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“…there exists a function u ∈ L 2 (0, T ; H 3 (0, L)) ∩ H 1 (0, T ; H 1 (0, L)) * * This is the approach used recently by Rosier and Zhang [32] to establish the exact boundary controllability of the nonlinear Schrödinger equation posed on a bounded domain Ω in R n .…”

Section: If There Exists a Nonempty Open Setmentioning

confidence: 99%

Control and stabilization of the Korteweg-de Vries equation: recent progresses

2009

Self Cite

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The study of the control and stabilization of the KdV equation began with the work of Russell and Zhang in late 1980s. Both exact control and stabilization problems have been intensively studied since then and significant progresses have been made due to many people's hard work and contributions. In this article, the authors intend to give an overall review of the results obtained so far in the study but with an emphasis on its recent progresses. A list of open problems is also provided for further investigation.

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Exact boundary controllability of the nonlinear Schrödinger equation

Cited by 52 publications

References 27 publications

Output feedback stabilization of the linearized Korteweg–de Vries equation with right endpoint controllers

Output feedback stabilization of the linearized Korteweg–de Vries equation with right endpoint controllers

Asymptotics for Optimal Design Problems for the Schrödinger Equation with a Potential

Control and stabilization of the Korteweg-de Vries equation: recent progresses

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