Abstract
This paper addresses the stability problem of nonlinear systems with variable-time impulses. By B-equivalence method, we shall show that under the well-selected conditions each solution of the considered systems will intersect each surface of discontinuity exactly once, and that the considered systems can be reduced to the fixed-time impulsive ones, which can be regarded as the comparison systems of the considered variable-time impulsive systems. Based on the stability theory of fixed-time impulsive systems, we propose a set of stability criteria for the variable-time impulsive systems. The theoretical results are illustrated by impulsive stabilization of Chua circuit.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T. Yang, Impulsive Control Theory, Springer-Verlag: Berlin, 2001.
V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Science, Singapore, 1989.
Z. H. Guan, J. Lam, and G. R. Chen, “On impulsive autoassociative neural networks,” Neural Networks, vol. 13, pp. 63–69, 2000. [click]
C. D. Li, G. Feng, and T. W. Huang, “On hybrid impulsive and switching neural networks,” IEEE Transactions on Systems, Man, and Cybernetrics-B, vol. 38, no. 6, pp. 1549–1560, 2008.
A. Arbi, C. Aouiti, F. Cherif, A. Touati, and A. M. Alimi, “Stability analysis of delayed Hopfield neural networks with impulses via inequality techniques,” Neurocomputing, vol. 158, pp. 281–294, 2015. [click]
R. Rakkiyappan, G. Velmurugan, and X. D. Li, “Complete stability analysis of complex-valued neural networks with time delays and impulses,” Neural Processing Letters, vol. 41, pp. 435–468, 2015. [click]
W. H. Chen, X. M. Lu, and W. X. Zheng, “Impulsive stabilization and impulsive synchronization of discrete-time delayed neural networks,” IEEE Transactions on Neural Networks and Learning Systems, vol. 26, no. 4, pp. 734–748, 2015.
Y. Zhang, “Stability of discrete-time Markovian jump delay systems with delayed impulses and partly unknown transition probabilities,” Nonlinear Dynamics, vol. 75, no. 1-2, pp. 101–111, 2014. [click]
Q. Song and J. Zhang, “Global exponential stability of impulsive Cohen-CGrossberg neural network with timevarying delays,” Nonlinear Analysis: Real World Applications, vol. 9, no. 2, pp. 500–510, 2008. [click]
S. J. Long and D. Y. Xu, “Delay-dependent stability analysis for impulsive neural networks with time varying delays,” Neurocomputing, vol. 71, no. 7-9, pp. 1705–1713, 2008. [click]
X. Li and S. Song, “Impulsive control for existence, uniqueness and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays,” IEEE Transactions on Neural Networks, vol. 24, pp. 868–877, 2013.
V. Lakshmikantham, S. Leela, and S. Kaul, “Comparison principle for impulsive differential equations with variable times and Stability theory,” Nonlinear Analalysis, vol. 22, no. 4, pp. 499–503, 1994. [click]
S. Kaul, V. Lakshmikantham, and S. Leela, “Extremal solutions, comparison principle and stability criteria for impulsive differential equations with variable times,” Nonlinear Analysis: Theory, Methods & Applications, vol. 22, no. 10, pp. 1263–1270, 1994. [click]
M. Frigon and D. O’Regan, “Impulsive differential equations with variable times,” Nonlinear Analalysis, vol. 26, no. 12, pp.1913–1922, 1996.
X. L. Fu, J. G. Qi, and Y. S. Liu, “General comparison principle for impulsive variable time differential equations with application,” Nonlinear Analysis, vol. 42, no. 8, pp. 1421–1429, 2000.
M. Akhmet, Principles of Discontinuous Dynamical Systems, Springer, New York, 2010.
E. Akalin and M. Akhmet, “The principles of B-smooth discontinuous flows,” Computers and Mathematics with Applications, vol. 49, pp. 981–995, 2005. [click]
M. Akhmet, “On the general problem of stability for impulsive differential equations,” Journal of Mathematical Analysis and Applications, vol. 288, pp. 182–196, 2003. [click]
M. Akhmet, “Perturbations and Hopf bifurcation of the planar discontinuous dynamical system,” Nonlinear Analysis, vol. 60, pp. 163–178, 2003. [click]
M. Akhmet, “Li-Yorkechaos in the system with impacts,” Journal of Mathematical Analysis and Applications, vol. 351, pp. 804–810, 2009. [click]
M. Akhmet and N. Perestyuk, “The comparison method for differential equations with impulse action,” Differential Equations, vol. 26, no. 9, pp. 1079–1086, 2009.
M. Akhmet and M. Turan., “Differential equations on variable time scales,” Nonlinear Analysis, vol. 70, pp. 1175–1192, 2009. [click]
M. Sayli and E. Yilmaz, “Global robust asymptotic stability of variable-time impulsive BAMneural networks,” Neural Networks, vol. 60, pp. 67–73, 2014. [click]
Y. Tang, H. J. Gao, W. Zhang, and J. Kurths, “Leaderfollowing consensus of a class of stochastic delayed multiagent systems with partial mixed impulses,” Automatica, 53, 346–354, 2015. [click]
X. Li, D. O’Regan, and H. Akca, “Global exponential stabilization of impulsive neural networks with unbounded continuously distributed delays,” IMA Journal of Applied Mathematics, vol. 80, no. 1, pp. 85–99, 2015. [click]
X. Li, M. Bohner, and C. K. Wang, “Impulsive differential equations: Periodic solutions and applications,” Automatica, vol. 52, pp.173–178, 2015.
H. M. Wang, S. K. Duan, C. D. Li, L. Wang, and T. Huang, “Stability criterion of linear delayed impulsive differential systems with impulse time windows,” International Journal of Control, Automation and Systems, vol. 14, no. 1, pp. 174–180, 2016. [click]
J. Tan, C. D. Li, and T. W. Huang, “Stability of impulsive systems with time window via comparison method,” International Journal of Control, Automation and Systems, vol. 13, Is. 6, pp. 1346–1350, 2015. [click]
Author information
Authors and Affiliations
Corresponding author
Additional information
Recommended by Associate Editor Ho Jae Lee under the direction of Editor Yoshito Ohta. This work is supported by Natural Science Foundation of China (Grant nos: 61403313, 61374078) and the work was partially supported by Research Foundation of Key laboratory of Machine Perception and Children’s Intelligence Development funded by CQUE, China. This publication was made possible by NPRP Grant No. NPRP 4-1162-1-181 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors.
Chuandong Li received his B.S. degree in Applied Mathematics from Sichuan University, Chengdu, China in 1992, and an M.S. degree in operational research and control theory and a Ph.D. degree in Computer Software and Theory from Chongqing University, Chongqing, China, in 2001 and 2005, respectively. He has been a professor at the College of Electronic and Information Engineering, Southwest University, Chongqing 400715, China, since 2012, and been the IEEE Senior member since 2010. From November 2006 to November 2008, he served as a research fellow in the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Hong Kong, China. He has published more than 100 journal papers. His current research interest covers computational intelligence, neural networks, memristive systems, chaos control and synchronization, and impulsive dynamical systems.
Yinghua Zhou received his B.S. degree in Communication Engineering from Chongqing University of Posts and Telecommunications, Chongqing, China in 2003, and his M.S. degree in Communication and information system from Chongqing University of Posts and Telecommunications, Chongqing, China, in 2006. He is currently pursuing a Ph.D. degree with College of Electronic and Information Engineering, Southwest University, Chongqing, China. His current research interests include neural networks, memristive systems, stability and synchronization of impulsive dynamics systems.
Hui Wang received her B.S. degree in Applied Mathematics from Chongqing Normal University, Chongqing, China in 1999, and an M.S. degree in operational research and control theory and a Ph.D. degree in Computer Software and Theory from Chongqing University, Chongqing, China, in 2003 and in 2007, respectively. He has been a professor at the College of Mathematics Theory, Chongqing Normal, Chongqing 400044, China, since 2014. She has published more than 30 journal papers. Her current research interest covers computational intelligence, memristive systems, chaos control and synchronization, and impulsive dynamical systems.
Tingwen Huang obtained his B.S. from Southwest Normal University in 1990, an M.S. from Sichuan University in 1993 and a Ph.D. from Texas A&M University in 2002. After he graduated at Texas A&M University, he has been working in Mathematics Department of Texas A&M University as Visiting Assistant Professor. In 2003, he started to work at Texas A&M University at Qatar until now. He now is an associate professor of Mathematics. His research fields include neural networks, chaos and its applications, etc. He has published about 30 journal papers on neural networks and nonlinear dynamics.
Rights and permissions
About this article
Cite this article
Li, C., Zhou, Y., Wang, H. et al. Stability of nonlinear systems with variable-time impulses: B-equivalence method. Int. J. Control Autom. Syst. 15, 2072–2079 (2017). https://doi.org/10.1007/s12555-016-0086-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12555-016-0086-7