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Asymptotic Lyapunov stability with probability one of quasi linear systems subject to multi-time-delayed feedback control and wide-band parametric random excitation

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Abstract

The asymptotic Lyapunov stability with probability one of multi-degree-of-freedom quasi linear systems subject to multi-time-delayed feedback control and multiplicative (parametric) excitation of wide-band random process is studied. First, the multi-time-delayed feedback control forces are expressed approximately in terms of the system state variables without time delay and the system is converted into ordinary quasi linear system. Then, the averaged Itô stochastic differential equations are derived by using the stochastic averaging method for quasi linear systems and the expression for the largest Lyapunov exponent of the linearized averaged Itô equations is derived. Finally, the necessary and sufficient condition for the asymptotic Lyapunov stability with probability one of the trivial solution of the original system is obtained approximately by letting the largest Lyapunov exponent to be negative. An example is worked out in detail to illustrate the application and validity of the proposed procedure and to show the effect of the time delay in feedback control on the largest Lyapunov exponent and the stability of system.

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Authors and Affiliations

  1. School of Civil Engineering and Transportation, South China University of Technology, 510640, Guangzhou, People’s Republic of China

    Xue Ping Li

  2. Department of Civil Engineering, Xiamen University, 361005, Xiamen, People’s Republic of China

    Zhong Hua Liu

  3. College of Civil Engineering and Architecture, Zhejiang University, 310027, Hangzhou, People’s Republic of China

    Rong Hua Huan

  4. Department of Mechanics, School of Aeronautics and Astronautics, Zhejiang University, 310027, Hangzhou, People’s Republic of China

    Wei Qiu Zhu

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  1. Xue Ping Li
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  2. Zhong Hua Liu
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  3. Rong Hua Huan
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  4. Wei Qiu Zhu
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Correspondence to Xue Ping Li.

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Li, X.P., Liu, Z.H., Huan, R.H. et al. Asymptotic Lyapunov stability with probability one of quasi linear systems subject to multi-time-delayed feedback control and wide-band parametric random excitation. Arch Appl Mech 79, 1051–1061 (2009). https://doi.org/10.1007/s00419-008-0273-y

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  • DOI: https://doi.org/10.1007/s00419-008-0273-y

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