Periodicity and stabilization control of the delayed Filippov system with perturbation

Zuowei Cai; Jianhua Huang; Liu Yang; Lihong Huang

doi:10.3934/dcdsb.2019235

Article Contents

2020, Volume 25, Issue 4: 1439-1467. Doi: 10.3934/dcdsb.2019235

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Periodicity and stabilization control of the delayed Filippov system with perturbation

a.
Hunan Provincial Key Laboratory of Intelligent Information Processing and Application, Hengyang, Hunan 421002, China
b.
School of Information Science and Engineering, Hunan Women's University, Changsha, Hunan 410002, China
c.
College of Science, National University of Defense Technology, Changsha, Hunan 410073, China
d.
School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan 410114, China

^* Corresponding author: Jianhua Huang
^* Corresponding author: Jianhua Huang

Received: July 31, 2018

Revised: April 30, 2019

Early access: November 2019

Published: April 2020

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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Abstract

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.

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

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Figure 1. Discontinuous activation functions of Example 1

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

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

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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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