Research ArticlesNo Access

Effects of variable-time impulses on global exponential stability of Cohen–Grossberg neural networks

Xianxiu Zhang

College of Electronic and Information Engineering, Southwest University, Chongqing 400715, P. R. China

Department of Mathematics, Liupanshui Normal University, Guizhou, Liupanshui 553001, P. R. China

E-mail Address: zxx1234567@sina.com

Search for more papers by this author

Chuandong Li

http://orcid.org/0000-0001-6155-4849

College of Electronic and Information Engineering, Southwest University, Chongqing 400715, P. R. China

E-mail Address: cdli@swu.edu.cn

Corresponding author.

Search for more papers by this author

Tingwen Huang

Department of Mathematics, Texas A&M University at Qatar, Doha 23874, Qatar

Search for more papers by this author

, and

Hafiz Gulfam Ahmad

Department of Computer Science and IT, Ghazi University, D. G. Khan 32260, Pakistan

Search for more papers by this author

https://doi.org/10.1142/S1793524517501170Cited by:5 (Source: Crossref)

Abstract

We investigate the global exponential stability of Cohen–Grossberg neural networks (CGNNs) with variable moments of impulses using B-equivalence method. Under certain conditions, we show that each solution of the considered system intersects each surface of discontinuity exactly once, and that the variable-time impulsive systems can be reduced to the fixed-time impulsive ones. The obtained results imply that impulsive CGNN will remain stability property of continuous subsystem even if the impulses are of somewhat destabilizing, and that stabilizing impulses can stabilize the unstable continuous subsystem at its equilibrium points. Moreover, two stability criteria for the considered CGNN by use of proposed comparison system are obtained. Finally, the theoretical results are illustrated by two examples.

Keywords:

AMSC: 34D23

Remember to check out the Most Cited Articles in IJB!

Check out new Biomathematics books in our Mathematics 2018 catalogue!
Featuring author Frederic Y M Wan and more!

References

1. M. A. Cohen and S. Grossberg , Absolute stability of global pattern formulation and parallel memory storage by competitive neural networks, IEEE Trans. Syst. Man Cybern. 13 (1983) 815–826. Crossref, Web of Science, Google Scholar
2. B. Liu and L. Huang , Existence and exponential stability of periodic solutions for a class of Cohen–Grossberg neural networks with time-varying delays, Chaos, Solitons Fractals 32 (2007) 617–627. Crossref, Web of Science, Google Scholar
3. Y. Li , Existence and exponential stability of periodic solutions for Cohen–Grossberg neural networks with multiple delays, Chaos, Solitons Fractals 20 (2004) 459–466. Crossref, Web of Science, Google Scholar
4. W. Zhao , Dynamics of Cohen–Grossberg neural network with variable coefficients and time-varying delays, Nonlinear Anal. 9 (2008) 1024–1037. Crossref, Web of Science, Google Scholar
5. C. Li and S. Yang , Existence and attractivity of periodic solutions to non-autonomous Cohen–Grossberg neural networks with time delays, Chaos, Solitons Fractals 41 (2009) 1235–1244. Crossref, Web of Science, Google Scholar
6. L. Wang and X. Zou , Exponential stability of Cohen–Grossberg neural network, Neural Netw. 15 (2002) 415–422. Crossref, Web of Science, Google Scholar
7. X. F. Liao, C. G. Li and K. W. Wong , Criteria for exponential stability of Cohen–Grossberg neural network, Neural Netw. 17 (2004) 1401–1414. Crossref, Web of Science, Google Scholar
8. H. Xiang and J. Cao , Exponential stability of periodic solutions to Cohen–Grossberg type BAM neural networks with time-varying delays, Neurocomputing 72 (2009) 1702–1711. Crossref, Web of Science, Google Scholar
9. X. Liao, J. Yang and S. Guo , Exponential stability of Cohen–Grossberg neural networks with delays, Commun. Nonlinear Sci. Numer. Simulat. 13 (2008) 1767–1775. Crossref, Web of Science, Google Scholar
10. W. Zhao , Global exponential stability analysis of Cohen–Grossberg neural networks with delays, Commun. Nonlinear Sci. Numer. Simulat. 13 (2008) 847–856. Crossref, Web of Science, Google Scholar
11. H. Zhang, L. S. Chen and P. Georgescu , Impulsive control strategies for pest management, J. Biol. Syst. 15(2) (2007) 235–260. Link, Web of Science, Google Scholar
12. H. Zhang, W. J. Xu and L. S. Chen , A impulsive infective transmission SI model for pest control, Math. Meth. Appl. Sci. 30(10) (2007) 1169–1184. Crossref, Web of Science, Google Scholar
13. Z. Zhao, L. S. Chen and X. Y. Song , Impulsive vaccination of SEIR epidemic model with time delay and nonlinear incidence rate, Math. Comput. Simulat. 79(3) (2008) 500–510. Crossref, Web of Science, Google Scholar
14. R. Korn , Some applications of impulse control in mathematical finance, Math. Meth. Oper. Res. 50(3) (1999) 493–518. Crossref, Web of Science, Google Scholar
15. C. J. Li, C. D. Li, X. F. Liao and T. W. Huang , Impulsive effects on stability of high-order BAM neural networks with time delays, Neurocomputing 74(10) (2011) 1541–1550. Crossref, Web of Science, Google Scholar
16. Z. H. Guan, J. Lam and G. R. Chen , On impulsive auto-associative neural networks, Neural Netw. 13 (2000) 63–69. Crossref, Web of Science, Google Scholar
17. Z. H. Guan and G. R. Chen , On delayed impulsive Hopfield neural networks, Neural Netw. 12 (1999) 273–280. Crossref, Web of Science, Google Scholar
18. W. H. Chen, X. M. Lu and W. X. Zheng , Impulsive stabilization and impulsive synchronization of discrete-time delayed neural networks, IEEE Trans. Neural Netw. Learn. Syst. 26(4) (2015) 734–748. Crossref, Web of Science, Google Scholar
19. Y. Zhang , Stability of discrete-time Markovian jump delay systems with delayed impulses and partly unknown transition probabilities, Nonlinear Dynam. 75(1–2) (2014) 101–111. Crossref, Web of Science, Google Scholar
20. 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 Trans. Neural Netw. 24 (2013) 868–877. Crossref, Web of Science, Google Scholar
21. R. Rakkiyappan, G. Velmurugan and X. D. Li , Complete stability analysis of complex-valued neural networks with time delays and impulses, Neural Process. Lett. 41 (2015) 435–468. Crossref, Web of Science, Google Scholar
22. T. Yang , Impulsive Control Theory (Springer-Verlag, Berlin, 2001). Google Scholar
23. V. Lakshmikantham, D. D. Bainov and P. S. Simeonov , Theory of Impulsive Differential Equations (World Scientific, Singapore, 1989). Link, Google Scholar
24. M. Sayli and E. Yilmaz , Global robust asymptotic stability of variable-time impulsive BAM neural networks, Neural Netw. 60 (2014) 67–73. Crossref, Web of Science, Google Scholar
25. M. Sayli and E. Yilmaz , State-dependent impulsive Cohen–Grossberg neural networks with time-varying delays, Neurocomputing 171 (2016) 1375–1386. Crossref, Web of Science, Google Scholar
26. S. Leela, V. Lakshmikantham and S. Kaul , Extremal solutions, comparison principle and stability criteria for impulsive differential equations with variable times, Nonlinear Anal. 22 (1994) 1263–1270. Crossref, Web of Science, Google Scholar
27. C. Liu, C. Li and X. Liao , Variable-time impulses in BAM neural networks with delay, Neurocomputing 74 (2011) 3286–3295. Crossref, Web of Science, Google Scholar
28. M. Akhmet , Principles of Discontinuous Dynamical Systems (Springer, New York, 2010). Crossref, Google Scholar
29. X. Wang, C. Li, T. Huang and L. Chen , Impulsive exponential synchronization of randomly coupled neural networks with Markovian jumping and mixed model-dependent time delays, Neural Netw. 60 (2014) 25–32. Crossref, Web of Science, Google Scholar
30. W. Zhang, C. Li, T. Huang and X. He , Synchronization of memristor-based coupling recurrent neural networks with time-varying delays and impulses, IEEE Trans. Neural Netw. Learn. Syst. 26(12) (2015) 3308–3313. Crossref, Web of Science, Google Scholar
31. P. Baldi and A. F. Atiya , How delays affect neural dynamics and learning, IEEE Trans. Neural Netw. 5 (1994) 612–621. Crossref, Web of Science, Google Scholar
32. J. L. Wang et al., Convergence behavior of delayed discrete cellular neural network without periodic coefficients, Neural Netw. 53 (2014) 61–68. Crossref, Web of Science, Google Scholar
33. J. de Cao and L. Wang , Exponential stability and periodic oscillatory solution in BAM networks with delays, IEEE Trans. Neural Netw. 13 (2002) 457–463. Crossref, Web of Science, Google Scholar