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finite-time input-to-state stability for Infinite-Dimensional Systems

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL 2025年第8期35卷 3141-3153页

作者： Sun, Xiaorong Zheng, Jun Zhu, Guchuan Southwest Jiaotong Univ Sch Math Chengdu Peoples R China Polytech Montreal Dept Elect Engn Montreal PQ Canada

In this paper, we extend the notion of finite-time input-to-state stability (FTISS) for finite-dimensional systems to infinite-dimensional systems. More specifically, we first prove an FTISS Lyapunov theorem for a class of infinite-dimensional systems, namely, the existence of an FTISS Lyapunov functional (FTISS-LF) implies the FTISS of the system, and then, provide a sufficient condition for ensuring the existence of an FTISS-LF for a class of abstract infinite-dimensional systems under the framework of compact semigroup theory and Hilbert spaces. As an application of the FTISS Lyapunov theorem, we verify the FTISS for a class of parabolic PDEs involving sublinear terms and distributed in-domain disturbances. Since the non-linear terms of the corresponding abstract system are not Lipschitz continuous, the well-posedness is proved based on the application of compact semigroup theory, and the FTISS is assessed by using the Lyapunov method with the aid of an interpolation inequality. Numerical simulations are conducted to confirm the theoretical results.

关键词： compact semigroup finite-time input-to-state stability infinite-dimensional system input-to-state stability interpolation inequality Lyapunov functional parabolic equation

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finite-time input-to-state stability of discrete-time stochastic switched systems: a comparison principle-based method

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INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE 2023年第1期54卷 1-16页

作者： Liang, Shuang Liang, Jinling Qiu, Jianlong Southeast Univ Sch Math Nanjing Peoples R China Linyi Univ Sch Automat & Elect Engn Linyi Shandong Peoples R China

This paper studies the input-to-state stability (ISS) and the finite-time input-to-state stability issues for the discrete-time stochastic switched nonlinear systems with time-varying delay. Based on the improved vector-version comparison principle and by employing the stochastic analysis method, sufficient conditions are first derived to guarantee the discrete-time stochastic switched system to be finite-time input-to-state stable in mean and finite-time stochastic input-to-state stable. Then, the comparison principle is further utilised to establish the p-th moment ISS result. As an application, the mean-square ISS is analysed for the linear discrete-time stochastic switched system. At the end of the paper, two numerical examples are provided to illustrate the feasibility/effectiveness of the obtained results.

关键词： finite-time input-to-state stability finite-time stochastic input-to-state stability comparison principle stochastic switched system

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finite-time input-to-state stability and settling-time estimation of impulsive switched systems with multiple impulses

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JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS 2024年第17期361卷

作者： Zhang, Taixiang Cao, Jinde Li, Xiaodi Hua, Liang Southeast Univ Sch Math Nanjing 211189 Peoples R China Shandong Normal Univ Sch Math & Stat Jinan 250014 Peoples R China Nantong Univ Sch Elect Engn Nantong 226019 Peoples R China

This paper investigates finite-time input-to-state stability (FT-ISS) of impulsive switched systems with multiple impulses. Some FT-ISS conditions, using Lyapunov method and dwell-time condition, are established for impulsive switched systems involving destabilizing and stabilizing impulses simultaneously. When constituent modes regulating continuous dynamics are FT-ISS and discrete dynamics involve destabilizing and stabilizing impulses, it is shown that, the FTISS is retained if impulsive-switching signal satisfies some dwell-time condition. When some constituent modes regulating continuous dynamics are not FT-ISS and discrete dynamics involve destabilizing and stabilizing impulses, it is shown that, the impulsive-switching signal which satisfies some dwell-time conditions can achieve the FT-ISS of system. In addition, the settling time can be derived conveniently for certain impulsive-switching signal that is formalized by dwell-time condition. The estimation of settling time presents a class of uniformity with respect to the impulsive-switching signals. Two examples are finally presented for the proposed FT-ISS results.

关键词： Impulsive switched systems Settling time Destabilizing impulses Stabilizing impulses finite-time input-to-state stability finite-time input-to-state stability Impulsive switched systems Settling time Destabilizing impulses Stabilizing impulses

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Sufficient conditions on finite-time input-to-state stability of nonlinear stochastic impulsive time-varying systems

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INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL 2023年第11期33卷 6417-6430页

作者： Zhang, Meng Zhu, Quanxin Shandong Univ Finance & Econ Sch Stat & Math Jinan 250014 Peoples R China Hunan Normal Univ Sch Math & Stat CHP LCOCS Changsha 410081 Peoples R China

The finite-time input-to-state stability (FTISS) of stochastic impulsive time-varying nonlinear systems is investigated in this paper. Different from the previous works concerning FTISS, we allow the coefficients of the estimated upper bound for the diffusion operator of a Lyapunov function to be time-varying. Moreover, a relaxed relation between the impulsive frequency is provided. Leveraging the Lyapunov method along with Lambert W function and average impulsive interval (AII) method, the input-to-state (ISS) property is obtained within finite time. Furthermore, two examples are provided to verify the validity of the conclusion.

关键词： finite-time input-to-state stability lambert W function Lyapunov method stochastic impulsive system

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finite-time input-to-state stability of switched stochastic time-varying nonlinear systems with time delays

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CHAOS SOLITONS & FRACTALS 2022年第0期162卷

作者： Zhang, Meng Zhu, Quanxin Shandong Univ Finance & Econ Sch Stat & Math Jinan 250014 Peoples R China Hunan Normal Univ Sch Math & Stat CHP LCOCS Changsha 410081 Peoples R China

This article extends the finite-time input-to-state stability (FTISS) to switched stochastic time-varying nonlinear systems with time delays. By using the methods of Razumikhin theorem, comparison principle combined with average dwell-time (ADT) approach, several sufficient conditions of nonlinear switched stochastic delayed sys-tems are analyzed to obtain ISS property in the framework of finite time. Compared with the existing results, we allow the upper bound estimation for the derivative of Lyapunov function to be time-varying, which takes both positive and negative values. Finally, an example is given to illustrate the effectiveness of the proposed results.(c) 2022 Elsevier Ltd. All rights reserved.

关键词： finite-time input-to-state stability Razumikhin theorem Switched stochastic system time delays

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Sufficient conditions on finite-time input-to-state stability of nonlinear impulsive systems: a relaxed Lyapunov function method

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INTERNATIONAL JOURNAL OF CONTROL 2022年第11期95卷 2992-3001页

作者： Wang, Zengyun Cao, Jinde Cai, Zuowei Hunan First Normal Univ Dept Math Changsha Hunan Peoples R China Southeast Univ Sch Math Nanjing 210096 Jiangsu Peoples R China Southeast Univ Jiangsu Prov Key Lab Networked Collect Intelligen Nanjing 210096 Jiangsu Peoples R China Yonsei Univ Yonsei Frontier Lab Seoul South Korea Hunan Womens Univ Dept Informat Technol Changsha Hunan Peoples R China

This article presents sufficient conditions on finite-time input-to-state stability (FTISS) of nonlinear impulsive systems, which gathers the analysis of input-to-state stability (ISS) and finite-time convergence rate. By utilising the finite-time Stable Function Pair (FTSFP) and Generalised KL (GKL) function, we established an unified FTISS criterion for nonlinear impulsive systems, in which the impulses maybe beneficial to stabilisation or detrimental to stabilisation. Moreover, a relaxed Lyapunov Function (LF) method is used to investigate FTISS. Compared with the traditional LF method, the main advantage is that the constructed LF is permitted to own indefinite-derivative. As a special case of FTISS, a corollary on finite-time stability (FTS) is established, which is applicable to investigate the realisation of FTS for nonlinear system via impulsive control strategy. Finally, the proposed results are supported by two practical numerical examples.

关键词： finite-time input-to-state stability nonlinear impulsive system finite-time stable function pair generalised KL function relaxed Lyapunov function method

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finite/fixed-time input-to-state stability of impulsive nonlinear systems

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APPLIED MATHEMATICAL MODELLING 2023年第1期123卷 23-38页

作者： Wei, Tengda Li, Xiaodi Shandong Normal Univ Sch Math & Stat Jinan 250014 Peoples R China Shandong Normal Univ Characterist Lab Control & Engn Computat Shandong Jinan 250014 Peoples R China

In this paper, we investigate the finite-time and fixed-time input-to-state stability of impulsive nonlinear systems involving hybrid impulses which have time-dependent multiple jump maps. Under unified average dwell-time conditions, several criteria for finite-time and fixed-time input-to-state stability are established by virtue of the Lyapunov function with linear rate and nonlinear rate of jump maps. It shows that, when the finite/fixed-time input-to-state stable continuous systems are subject to hybrid impulses, the finite-time and fixed-time input-to-state stability can be ascertained under the requirement that the accumulative effect of hybrid impulses is non-destabilizing or the accumulative effect of hybrid impulses is destabilizing and the impulses are not activated too frequently. Finally, the effectiveness of theoretical results is verified by two numerical examples and an application example about an axisymmetric circular spacecraft. & COPY;2023 Elsevier Inc. All rights reserved.

关键词： Hybrid impulse finite-time input-to-state stability Fixed-time input-to-state stability Lyapunov method Dwell-time condition

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finite-time stochastic input-to-state stability of switched stochastic nonlinear systems

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APPLIED MATHEMATICS AND COMPUTATION 2015年 268卷 1038-1054页

作者： Zhao, Ping Feng, Wei Zhao, Yan Kang, Yu Univ Jinan Sch Elect Engn Jinan 250022 Shandong Peoples R China Shandong Agr Univ Coll Informat Sci & Engn Tai An 271018 Shandong Peoples R China Univ Jinan Sch Math Sci Jinan 250022 Shandong Peoples R China Univ Sci & Technol China Dept Automat Hefei 230026 Anhui Peoples R China

In this paper, the problems of finite-time globally asymptotical stability in probability (FGSP) and finite-time stochastic input-to-state stability (FSISS) for switched stochastic nonlinear (SSNL) systems are investigated. To solve these problems elegantly, some new definitions on FGSP and FSISS are presented in the form of generalized KL(GKL) function, and some lemmas about GKL. functions and their properties are proved. Based on that, some sufficient conditions are provided firstly for nonswitched stochastic nonlinear (nSSNL) systems, which will make the corresponding study on SSNL systems easier. Then, overcoming the difficulties coming with the appearance of switching, some sufficient conditions on FGSP and FSISS are given for SSNL systems. Moreover, based on the concept of average dwell-time, a sufficient condition for FSISS of SSNL systems is also provided. Finally, some simulation examples are given to demonstrate the effectiveness of our results. (C) 2015 Elsevier Inc. All rights reserved.

关键词： Stochastic nonlinear system Switched system finite-time input-to-state stability finite-time globally asymptotically stable in probability FSISS-Lyapunov function

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A Lyapunov-Based Small-Gain Theorem for Interconnected finite-time input-to-state Stable Systems 22

A Lyapunov-Based Small-Gain Theorem for Interconnected Finit...

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22nd World Congress of the International Federation of Automatic Control (IFAC)

作者： Li, Huanhuan Liu, Tengfei Jiang, Zhong-Ping Northeastern Univ State Key Lab Synthet Automat Proc Ind Shenyang 110819 Peoples R China NYU Dept Elect & Comp Engn 370 Jay St Brooklyn NY 11201 USA

ISBN: (纸本)9781713872344

This paper presents a Lyapunov formulation of the small-gain theorem for the finite-time input-to-state stability (FTISS) of an interconnected system composed of FTISS subsystems. In addition, an FTISS-Lyapunov function for the entire system is constructed from the FTISS-Lyapunov functions of the subsystems. With respect to the previously developed nonlinear, Lyapunov-based small-gain theorem restricted to input-to-state stability, a new power-function-based scaling technique is proposed to deal with the challenge that a nonlinearly scaled FTISS-Lyapunov function may not retain a decreasing rate as a power function with a positive power less than one. Copyright (c) 2023 The Authors. This is an open access article under the CC BY-NC-ND license (https://***/licenses/by-nc-nd/4.0/)

关键词： finite-time input-to-state stability small-gain theorem Lyapunov functions

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Practical Event-Triggered finite-time Second-Order Sliding Mode Controller Design

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IEEE TRANSACTIONS ON CYBERNETICS 2024年第3期54卷 1972-1983页

作者： Dou, Wenhui Ding, Shihong Park, Ju H. Jiangsu Univ Sch Elect Informat Engn Zhenjiang 212013 Peoples R China Yeungnam Univ Dept Elect Engn Gyongsan 38541 South Korea

This article proposes a novel event-triggered second-order sliding mode (SOSM) control algorithm using the small-gain theorems. The developed algorithm has global event property in aspects of the triggering time intervals. First, an SOSM controller is designed related to the sampling error of states, and it is proved that the closed-loop system is finite-time input-to-state stable (FTISS) with the sampling error via utilizing the small-gain theorems. Second, combined with the constructed SOSM controller, a new triggering mechanism is proposed depending on the sampling error by designing the appropriate FTISS gain condition. Third, the practical finite-time stability of the closed-loop system is verified. It is shown that the minimum triggering time interval is always a positive value in the whole state space. Finally, the simulation results demonstrate the effectiveness of the developed control method.

关键词： Control systems Nonlinear systems Asymptotic stability Switches stability criteria Convergence Technological innovation Event-triggered control (ETC) finite-time input-to-state stability small-gain theorems second-order sliding mode (SOSM)

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