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A New Slack Lyapunov Functional for Dynamical System with Time Delay

Author

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  • Can Zhao

    (School of Information Science and Engineering, Chengdu University, Chengdu 610106, China
    Institute of Electronic and Information Engineering of UESTC in Guangdong, Dongguan 523808, China)

  • Kaibo Shi

    (School of Information Science and Engineering, Chengdu University, Chengdu 610106, China)

  • Yiqian Tang

    (School of Information Science and Engineering, Chengdu University, Chengdu 610106, China)

  • Shouming Zhong

    (School of Mathematics Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China)

Abstract

The traditional method of constructing a Lyapunov functional for dynamical systems with time delay is usually dependent on positive definite matrices in the quadratic form. In this paper, a new Lyapunov functional is proposed and the corresponding proof is given. It do not require that all matrices in the quadratic form of Lyapunov functionals are positive definite, while the quadratic form is still positive definite, which makes the estimate more relaxed due to special construction of matrices. It is a general form and can be used in the performance analysis of a variety of dynamical systems. Moreover, a lemma concerning the quadratic function is applied to deal with the quadratic term of time-varying delay. Lastly, in the case of classical dynamical systems with time delay, the verification results are given to illustrate the usefulness of the new slack Lyapunov functional.

Suggested Citation

  • Can Zhao & Kaibo Shi & Yiqian Tang & Shouming Zhong, 2022. "A New Slack Lyapunov Functional for Dynamical System with Time Delay," Mathematics, MDPI, vol. 10(23), pages 1-11, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:23:p:4462-:d:984783
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    References listed on IDEAS

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    1. de Oliveira, Fúlvia S.S. & Souza, Fernando O., 2020. "Further refinements in stability conditions for time-varying delay systems," Applied Mathematics and Computation, Elsevier, vol. 369(C).
    2. Shi, Kaibo & Cai, Xiao & She, Kun & Zhong, Shouming & Soh, YengChai & Kwon, OhMin, 2022. "Quantized memory proportional–integral control of active power sharing and frequency regulation in island microgrid under abnormal cyber attacks," Applied Energy, Elsevier, vol. 322(C).
    3. Zeng, Hong-Bing & Liu, Xiao-Gui & Wang, Wei, 2019. "A generalized free-matrix-based integral inequality for stability analysis of time-varying delay systems," Applied Mathematics and Computation, Elsevier, vol. 354(C), pages 1-8.
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    Cited by:

    1. Xinsong Yang & Ruofeng Rao, 2023. "Well-Posedness, Dynamics, and Control of Nonlinear Differential System with Initial-Boundary Value," Mathematics, MDPI, vol. 11(10), pages 1-4, May.

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