# American Institute of Mathematical Sciences

• Previous Article
A fractional-order delay differential model with optimal control for cancer treatment based on synergy between anti-angiogenic and immune cell therapies
• DCDS-S Home
• This Issue
• Next Article
Convergence analysis of the discrete duality finite volume scheme for the regularised Heston model

## 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, 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.   Google Scholar [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.  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. 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.  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. 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.  Google Scholar

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.   Google Scholar [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.  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. 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.  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. 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.  Google Scholar
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
 [1] Chuandong Li, Fali Ma, Tingwen Huang. 2-D analysis based iterative learning control for linear discrete-time systems with time delay. Journal of Industrial & Management Optimization, 2011, 7 (1) : 175-181. doi: 10.3934/jimo.2011.7.175 [2] Huan Su, Pengfei Wang, Xiaohua Ding. Stability analysis for discrete-time coupled systems with multi-diffusion by graph-theoretic approach and its application. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 253-269. doi: 10.3934/dcdsb.2016.21.253 [3] Michael Basin, Pablo Rodriguez-Ramirez. An optimal impulsive control regulator for linear systems. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 275-282. doi: 10.3934/naco.2011.1.275 [4] Alberto Bressan. Impulsive control of Lagrangian systems and locomotion in fluids. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 1-35. doi: 10.3934/dcds.2008.20.1 [5] Xueyan Yang, Xiaodi Li, Qiang Xi, Peiyong Duan. Review of stability and stabilization for impulsive delayed systems. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1495-1515. doi: 10.3934/mbe.2018069 [6] Vladimir Răsvan. On the central stability zone for linear discrete-time Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 734-741. doi: 10.3934/proc.2003.2003.734 [7] Xiang Xie, Honglei Xu, Xinming Cheng, Yilun Yu. Improved results on exponential stability of discrete-time switched delay systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 199-208. doi: 10.3934/dcdsb.2017010 [8] Pavel Drábek, Martina Langerová. Impulsive control of conservative periodic equations and systems: Variational approach. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3789-3802. doi: 10.3934/dcds.2018164 [9] Honglei Xu, Kok Lay Teo, Weihua Gui. Necessary and sufficient conditions for stability of impulsive switched linear systems. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1185-1195. doi: 10.3934/dcdsb.2011.16.1185 [10] Everaldo de Mello Bonotto, Daniela Paula Demuner. Stability and forward attractors for non-autonomous impulsive semidynamical systems. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1979-1996. doi: 10.3934/cpaa.2020087 [11] Ran Dong, Xuerong Mao. Asymptotic stabilization of continuous-time periodic stochastic systems by feedback control based on periodic discrete-time observations. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2020017 [12] C.Z. Wu, K. L. Teo. Global impulsive optimal control computation. Journal of Industrial & Management Optimization, 2006, 2 (4) : 435-450. doi: 10.3934/jimo.2006.2.435 [13] Byungik Kahng, Miguel Mendes. The characterization of maximal invariant sets of non-linear discrete-time control dynamical systems. Conference Publications, 2013, 2013 (special) : 393-406. doi: 10.3934/proc.2013.2013.393 [14] Hongyan Yan, Yun Sun, Yuanguo Zhu. A linear-quadratic control problem of uncertain discrete-time switched systems. Journal of Industrial & Management Optimization, 2017, 13 (1) : 267-282. doi: 10.3934/jimo.2016016 [15] Elena K. Kostousova. On control synthesis for uncertain dynamical discrete-time systems through polyhedral techniques. Conference Publications, 2015, 2015 (special) : 723-732. doi: 10.3934/proc.2015.0723 [16] Yuefen Chen, Yuanguo Zhu. Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems. Journal of Industrial & Management Optimization, 2018, 14 (3) : 913-930. doi: 10.3934/jimo.2017082 [17] Elena K. Kostousova. On polyhedral control synthesis for dynamical discrete-time systems under uncertainties and state constraints. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6149-6162. doi: 10.3934/dcds.2018153 [18] Victor Kozyakin. Minimax joint spectral radius and stabilizability of discrete-time linear switching control systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3537-3556. doi: 10.3934/dcdsb.2018277 [19] Yuyun Zhao, Yi Zhang, Tao Xu, Ling Bai, Qian Zhang. pth moment exponential stability of hybrid stochastic functional differential equations by feedback control based on discrete-time state observations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 209-226. doi: 10.3934/dcdsb.2017011 [20] Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2020020

2018 Impact Factor: 0.545