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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 and Continuous Dynamical Systems - S, 2020, 13 (6) : 1803-1811. doi: 10.3934/dcdss.2020106
##### References:
 [1] S. Ishi, K. Fukumizu and S. Watanabe, A network of chaotic elements for information processing, Neural Networks, 1 (1996), 25-40. [2] A. Khadra, X. 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. [3] V. Lakshmikantham, D. D. Bainoov and P. S. Simeonov, Theory of Impulsive Differential Equations, Singapore: World Scientific, 1989. doi: 10.1142/0906. [4] B. Liu and X. Liu, Robust stability of uncertain discrete impulsive systemss, IEEE Trans. Circuit Syst. II, Exp. Brief, 54 (2007), 455-459. [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. [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. [7] X. Xie, H. Xu, X. 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. [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. [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. [10] T. Yang, Impulsive Systems and Control: Theory and Applications, Huntington, New York: Nova Science Publishers, Inc., 2001. [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. [12] T. Yang, L. Yang and C. Yang, Impulsive control of Lorenz system, Physica D, 110 (1997), 18-24.  doi: 10.1016/S0167-2789(97)00116-4.

show all references

##### References:
 [1] S. Ishi, K. Fukumizu and S. Watanabe, A network of chaotic elements for information processing, Neural Networks, 1 (1996), 25-40. [2] A. Khadra, X. 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. [3] V. Lakshmikantham, D. D. Bainoov and P. S. Simeonov, Theory of Impulsive Differential Equations, Singapore: World Scientific, 1989. doi: 10.1142/0906. [4] B. Liu and X. Liu, Robust stability of uncertain discrete impulsive systemss, IEEE Trans. Circuit Syst. II, Exp. Brief, 54 (2007), 455-459. [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. [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. [7] X. Xie, H. Xu, X. 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. [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. [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. [10] T. Yang, Impulsive Systems and Control: Theory and Applications, Huntington, New York: Nova Science Publishers, Inc., 2001. [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. [12] T. Yang, L. Yang and C. Yang, Impulsive control of Lorenz system, Physica D, 110 (1997), 18-24.  doi: 10.1016/S0167-2789(97)00116-4.
State trajectory of $x_1(m)$ without nonlinear impulsive control
State trajectory of $x_2(m)$ without nonlinear impulsive control
State trajectory of $x_1(m)$ under nonlinear impulsive control
State trajectory of $x_2(m)$ under nonlinear impulsive control
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