Existence for Korteweg-de Vries-type equation with delay

Zhao, Zhihong; Rong, Erhua; Zhao, Xiukuan

doi:10.1186/1687-1847-2012-64

Cited by 6 publications

(1 citation statement)

References 18 publications

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“…We refer to [33] for the existence of the Korteweg-de Vries-type equation with delay and to the recent work [13] for the distributed stabilization of the Korteweg-de Vries-Burgers equation in the presence of input delay. Concerning the stabilization of the KdV equation by the construction of time varying feedback laws, without delay, we can cite the recent papers [10] and [31].…”

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confidence: 99%

On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback

Valein

2022

MCRF

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<p style='text-indent:20px;'>The aim of this work is to study the asymptotic stability of the nonlinear Korteweg-de Vries equation in the presence of a delayed term in the internal feedback. We first consider the case where the weight of the term with delay is smaller than the weight of the term without delay and we prove a semiglobal stability result for any lengths. Secondly we study the case where the support of the term without delay is not included in the support of the term with delay. In that case, we give a local exponential stability result if the weight of the delayed term is small enough. We illustrate these results by some numerical simulations.</p>

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confidence: 99%

On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback

Valein

2022

MCRF

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Planning of the LTE‐Advanced Radio Network

Penttinen¹

2015

The LTE‐Advanced Deployment Handbook

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Measurement of Freeform Optical Surfaces: Trade‐Off between Accuracy and Dynamic Range

Chen

Xue

Zhai

et al. 2020

Laser & Photonics Reviews

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Freeform optical surfaces are advantageous to optical designers, as they provide additional degrees of freedom for optimization. The loss of rotational symmetry, however, makes measurement of freeforms more challenging. Herein, advances in interferometric areal measurement of freeform optical surfaces, including null tests, non-null tests, near-null tests, and adaptive null tests, are reviewed. Some new developments, in the Hartmann test, deflectometry, and phase retrieval, as representatives of non-interferometric areal measurement, are then presented. Overall, the focus is on single-point-probe-based profilometry categorized into coordinate measurements and slope-or curvature-based measurements. Innovative measurement technology is trying to bridge the gap between high accuracy and high dynamic range as freeform optical surfaces are finding more applications in visible and shorter-wavelength optics.

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Existence for Korteweg-de Vries-type equation with delay

Abstract: The existence and uniqueness of the Korteweg-de Vries type equation with time delay are investigated. The problem is formulated as an abstract quasi-linear function differential equation. The solution is studied as a fixed point of the abstract integral equation.

Cited by 6 publications

References 18 publications

On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback

On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback

Planning of the LTE‐Advanced Radio Network

Measurement of Freeform Optical Surfaces: Trade‐Off between Accuracy and Dynamic Range

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