具有状态延迟的线性化Korteweg-de Vries方程的全局指数镇定

IF 1.6 4区 计算机科学 Q3 AUTOMATION & CONTROL SYSTEMS IMA Journal of Mathematical Control and Information Pub Date : 2023-05-30 DOI:10.1093/imamci/dnad016
Habib Ayadi, Mariem Jlassi
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引用次数: 0

摘要

摘要研究了一类具有状态时滞的一维线性Korteweg-de Vries方程(KdV)的适定性和全局边界指数镇定问题,该方程在有界区间$[0,2\pi]$上被设定,并在左边界被Dirichlet条件驱动。基于无延迟情况下的无限维反演方法,将系统进行线性volterra型积分变换,映射到另一个齐次目标系统,得到了显式反馈控制律。在此反馈下,利用适当的Lyapunov-Razumikhin泛函证明了所考虑的系统在适当的Banach空间上的适定性及其在$L^{2}(0,2\pi)$-范数拓扑上的指数镇定性。此外,在相同的反馈律下,我们得到了非线性KdV方程的局部指数稳定性。最后给出了数值算例。
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Global exponential stabilization of the linearized Korteweg-de Vries equation with a state delay
Abstract In this paper, well-posedness and global boundary exponential stabilization problems are studied for the one-dimensional linearized Korteweg-de Vries equation (KdV) with state delay, which is posed in bounded interval $[0,2\pi ]$ and actuated at the left boundary by Dirichlet condition. Based on the infinite-dimensional backstepping method for the delay-free case, a linear Volterra-type integral transformation maps the system into another homogeneous target system, and an explicit feedback control law is obtained. Under this feedback, we prove the well-posedness of the considered system in an appropriate Banach space and its exponential stabilization in the topology of $L^{2}(0,2\pi )$-norm by the use of an appropriate Lyapunov–Razumikhin functional. Moreover, under the same feedback law, we get the local exponential stability for the non-linear KdV equation. A numerical example is provided to illustrate the result.
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来源期刊
CiteScore
3.30
自引率
6.70%
发文量
33
审稿时长
>12 weeks
期刊介绍: The Journal is to provide an outlet for papers which are original and of high quality in mathematical control theory, systems theory, and applied information sciences. Short papers and mathematical correspondence or technical notes will be welcome, although the primary function of the journal is to publish papers of substantial length and coverage. The emphasis will be upon relevance, originality and clarify of presentation, although timeliness may well be an important feature in acceptable papers. Speculative papers that suggest new avenues for research or potential solutions to unsolved problems of control and information theory will be particularly welcome. Specific application papers will not normally be within the remit of the journal. Applications that illustrate techniques or theories will be acceptable. A prime function of the journal is to encourage the interplay between control and information theory and other mathematical sciences. All submitted papers will be judged on their merits by at least two referees and a full paper report will be available to the intending authors. Submitted articles will in general be published in an issue within six months of submission. Papers should not have previously published, nor should they be undes consideration for publication in another journal. This Journal takes publication ethics very seriously. If misconduct is found or suspected after the manuscript is published, the journal will investigate the matter and this may result in the article subsequently being retracted.
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