Generalized Lyapunov functionals for the input-to-state stability of infinite-dimensional systems

IF 4.8 2区计算机科学 Q1 AUTOMATION & CONTROL SYSTEMS

Automatica Pub Date : 2024-12-03 DOI:10.1016/j.automatica.2024.112005

Jun Zheng , Guchuan Zhu

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引用次数: 0

Abstract

This paper addresses the input-to-state stability (ISS) of infinite-dimensional systems by introducing a novel notion named generalized ISS-Lyapunov functional (GISS-LF) and the corresponding ISS Lyapunov theorem. Unlike the classical ISS-Lyapunov functional (ISS-LF) that must be positive definite, a GISS-LF can be positive semidefinite. Moreover, such a functional considers not only the relationship with elements in the state space but also takes into account the elements in the input space via a family of certain functionals. Consequently, this notion provides more options in constructing Lyapunov functionals for the ISS assessment of infinite-dimensional systems. In particular, we provide a positive answer to the open question raised by A. Mironchenko and C. Prieur, “Input-to-state stability of infinite-dimensional systems: recent results and open questions”, (Mironchenko and Prieur, 2020), regarding the existence of a coercive ISS-LF for the heat equation with Dirichlet boundary disturbances. To demonstrate the application of the proposed method, which we refer to as the generalized Lyapunov method, we present two examples, showing how to construct GISS-LFs by using positive semidefinite and non-coercive functionals for nonlinear parabolic equations defined over higher dimensional domains with Dirichlet boundary disturbances, and to derive small-gain conditions for guaranteeing the ISS with respect to distributed in-domain disturbances for coupled nonlinear degenerate parabolic equations, which contain ordinary differential equations as special cases.

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无限维系统输入到状态稳定性的广义Lyapunov泛函

通过引入广义ISS-Lyapunov泛函（GISS-LF）及其相应的ISS Lyapunov定理，研究了无限维系统的输入到状态稳定性（ISS）问题。与经典的ISS-Lyapunov泛函（ISS-LF）必须是正定的不同，GISS-LF可以是正半定的。此外，这种泛函不仅考虑与状态空间中元素的关系，而且还通过一定的泛函族考虑了输入空间中的元素。因此，这一概念为构建李雅普诺夫函数提供了更多的选择，用于ISS对无限维系统的评估。特别是，我们为a . Mironchenko和C. Prieur提出的开放问题提供了积极的答案，“无限维系统的输入到状态稳定性：最近的结果和开放的问题”，（Mironchenko和Prieur， 2020），关于具有Dirichlet边界扰动的热方程的强制ss - lf的存在性。为了演示所提出的方法的应用，我们将其称为广义Lyapunov方法，我们给出了两个例子，展示了如何使用正半定和非强制泛函来构造具有Dirichlet边界扰动的高维域上定义的非线性抛物方程的gsis - lfs。并推导了包含常微分方程的耦合非线性退化抛物方程在面对分布域内扰动时保证ISS的小增益条件。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Automatica 工程技术-工程：电子与电气

CiteScore

10.70

自引率

7.80%

发文量

617

审稿时长

5 months

期刊介绍： Automatica is a leading archival publication in the field of systems and control. The field encompasses today a broad set of areas and topics, and is thriving not only within itself but also in terms of its impact on other fields, such as communications, computers, biology, energy and economics. Since its inception in 1963, Automatica has kept abreast with the evolution of the field over the years, and has emerged as a leading publication driving the trends in the field. After being founded in 1963, Automatica became a journal of the International Federation of Automatic Control (IFAC) in 1969. It features a characteristic blend of theoretical and applied papers of archival, lasting value, reporting cutting edge research results by authors across the globe. It features articles in distinct categories, including regular, brief and survey papers, technical communiqués, correspondence items, as well as reviews on published books of interest to the readership. It occasionally publishes special issues on emerging new topics or established mature topics of interest to a broad audience. Automatica solicits original high-quality contributions in all the categories listed above, and in all areas of systems and control interpreted in a broad sense and evolving constantly. They may be submitted directly to a subject editor or to the Editor-in-Chief if not sure about the subject area. Editorial procedures in place assure careful, fair, and prompt handling of all submitted articles. Accepted papers appear in the journal in the shortest time feasible given production time constraints.