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Periodicity and stabilization control of the delayed Filippov system with perturbation

  • * Corresponding author: Jianhua Huang

    * Corresponding author: Jianhua Huang 
The first author is supported by National Natural Science Foundation of China (No.11701172, No.11771449), China Postdoctoral Science Foundation (No.2017M613361), the Science and Technology Plan Project of Hunan Province (No.2016TP1020), Open fund project of Hunan Provincial Key Laboratory of Intelligent Information Processing and Application for Hengyang normal university(No.2017IIPAZD02), Teaching Reform Project of Ordinary Colleges and Universities in Hunan Province (No. 911)
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  • By employing Leray-Schauder alternative theorem of set-valued maps and non-Lyapunov method (non-smooth analysis, inequality analysis, matrix theory), this paper investigates the problems of periodicity and stabilization for time-delayed Filippov system with perturbation. Several criteria are obtained to ensure the existence of periodic solution of time-delayed Filippov system by using differential inclusion. By designing appropriate switching state-feedback controller, the asymptotic stabilization and exponential stabilization control of Filippov system are realized. Applying these criteria and control design method to a class of time-delayed neural networks with perturbation and discontinuous activation functions under a periodic environment. The developed theorems improve the existing results and their effectiveness are demonstrated by numerical example.

    Mathematics Subject Classification: Primary: 39A23, 34D20, 34K09; Secondary: 34K20.


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  • Figure 1.  Discontinuous activation functions of Example 1

    Figure 2.  Time-domain behaviors of the state variables $ x_{1}(t) $ and $ x_{2}(t) $ of system (75) in Example 1

    Figure 3.  Time evolution of variables $ x_{1}(t) $ for system (75) under the switching state-feedback controller (48) in Example 1

    Figure 4.  Time evolution of variables $ x_{2}(t) $ for system (75) under the switching state-feedback controller (48) in Example 1

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