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June  2020, 13(6): 1803-1811. doi: 10.3934/dcdss.2020106

Stabilization of a discrete-time system via nonlinear impulsive control

1. 

School of Software Engineering, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

2. 

School of Information Science and Engineering, Fujian University of Technology, Fuzhou, Fujian 350118, China

3. 

Business School, Hunan Normal University, Changsha 410081, China

* Corresponding author: Jing Huang

Received  March 2018 Revised  August 2018 Published  September 2019

An impulsive control is one of the important stabilizing control strategies and exhibits many strong system performances such as shorten action time, low power consumption, effective resistance to uncertainty. This paper develops a nonlinear impulsive control approach to stabilize discrete-time dynamical systems. Sufficient conditions for asymptotical stability of discrete-time impulsively controlled systems are derived. Furthermore, an Ishi chaotic neural network is effectively stabilized by a designed nonlinear impulsive control.

Citation: Shaohong Fang, Jing Huang, Jinying Ma. Stabilization of a discrete-time system via nonlinear impulsive control. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1803-1811. doi: 10.3934/dcdss.2020106
References:
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X. Xie, H. Xu and R. Zhang, Exponential stabilization of impulsive switched systems with time delays using guaranteed cost control, Abstract and Applied Analysi, 2014 (2014), Art. ID 126836, 8 pp. doi: 10.1155/2014/126836.  Google Scholar

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H. Xu and K. L. Teo, Stabilizability of discrete chaotic systems via unified impulsive control, Physics Letters A, 374 (2009), 235-240.  doi: 10.1016/j.physleta.2009.10.065.  Google Scholar

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T. Yang, Impulsive Systems and Control: Theory and Applications, Huntington, New York: Nova Science Publishers, Inc., 2001. Google Scholar

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T. Yang and L. O. Chua, Impulsive stabilization for control and synchronization of chaotic systems: Theory and application to secure communication, IEEE Trans. Circuit Syst. I, Fundam. Theory Appl., 44 (1997), 976-988.  doi: 10.1109/81.633887.  Google Scholar

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show all references

References:
[1]

S. IshiK. Fukumizu and S. Watanabe, A network of chaotic elements for information processing, Neural Networks, 1 (1996), 25-40.   Google Scholar

[2]

A. KhadraX. Z. Liu and X. Shen, Synchronizing chaotic systems with delay and applications to secure communication, Automatica, 41 (2005), 1491-1502.  doi: 10.1016/j.automatica.2005.04.012.  Google Scholar

[3]

V. Lakshmikantham, D. D. Bainoov and P. S. Simeonov, Theory of Impulsive Differential Equations, Singapore: World Scientific, 1989. doi: 10.1142/0906.  Google Scholar

[4]

B. Liu and X. Liu, Robust stability of uncertain discrete impulsive systemss, IEEE Trans. Circuit Syst. II, Exp. Brief, 54 (2007), 455-459.   Google Scholar

[5]

X. Liu and K. L. Teo, Impulsive control of chaotic system, International Journal of Bifurcation and Chaos, 12 (2002), 1181-1190.  doi: 10.1142/S0218127402005029.  Google Scholar

[6]

T. Ushio, Chaotic synchronization and controlling chaos based on constraction mapping, Physics Letters A, 198 (1995), 14-22.  doi: 10.1016/0375-9601(94)01015-M.  Google Scholar

[7]

X. XieH. XuX. Cheng and Y. Yu, Improved results on exponential stability of discrete-time switched delay systems, Discrete & Continuous Dynamical Systems-Series B, 22 (2017), 199-208.  doi: 10.3934/dcdsb.2017010.  Google Scholar

[8]

X. Xie, H. Xu and R. Zhang, Exponential stabilization of impulsive switched systems with time delays using guaranteed cost control, Abstract and Applied Analysi, 2014 (2014), Art. ID 126836, 8 pp. doi: 10.1155/2014/126836.  Google Scholar

[9]

H. Xu and K. L. Teo, Stabilizability of discrete chaotic systems via unified impulsive control, Physics Letters A, 374 (2009), 235-240.  doi: 10.1016/j.physleta.2009.10.065.  Google Scholar

[10]

T. Yang, Impulsive Systems and Control: Theory and Applications, Huntington, New York: Nova Science Publishers, Inc., 2001. Google Scholar

[11]

T. Yang and L. O. Chua, Impulsive stabilization for control and synchronization of chaotic systems: Theory and application to secure communication, IEEE Trans. Circuit Syst. I, Fundam. Theory Appl., 44 (1997), 976-988.  doi: 10.1109/81.633887.  Google Scholar

[12]

T. YangL. Yang and C. Yang, Impulsive control of Lorenz system, Physica D, 110 (1997), 18-24.  doi: 10.1016/S0167-2789(97)00116-4.  Google Scholar

Figure 1.  State trajectory of $ x_1(m) $ without nonlinear impulsive control
Figure 2.  State trajectory of $ x_2(m) $ without nonlinear impulsive control
Figure 3.  State trajectory of $ x_1(m) $ under nonlinear impulsive control
Figure 4.  State trajectory of $ x_2(m) $ under nonlinear impulsive control
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