Journal of Applied Mathematics

Global Robust Attractive and Invariant Sets of Fuzzy Neural Networks with Delays and Impulses

Kaihong Zhao, Liwenjing Wang, and Juqing Liu

Full-text: Open access

Abstract

A class of fuzzy neural networks (FNNs) with time-varying delays and impulses is investigated. With removing some restrictions on the amplification functions, a new differential inequality is established, which improves previouse criteria. Applying this differential inequality, a series of new and useful criteria are obtained to ensure the existence of global robust attracting and invariant sets for FNNs with time-varying delays and impulses. Our main results allow much broader application for fuzzy and impulsive neural networks with or without delays. An example is given to illustrate the effectiveness of our results.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 935491, 8 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394807898

Digital Object Identifier
doi:10.1155/2013/935491

Zentralblatt MATH identifier
1266.92006

Citation

Zhao, Kaihong; Wang, Liwenjing; Liu, Juqing. Global Robust Attractive and Invariant Sets of Fuzzy Neural Networks with Delays and Impulses. J. Appl. Math. 2013 (2013), Article ID 935491, 8 pages. doi:10.1155/2013/935491. https://projecteuclid.org/euclid.jam/1394807898


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References

  • J. J. Hopfield, “Neurons with graded response have collective computational properties like those of two-state neurons,” Proceedings of the National Academy of Sciences of the United States of America, vol. 81, no. 10, pp. 3088–3092, 1984.
  • B. Kosko, “Bidirectional associative memories,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 18, no. 1, pp. 49–60, 1988.
  • L. O. Chua and L. Yang, “Cellular neural networks: theory,” IEEE Transactions on Circuits and Systems, vol. 35, no. 10, pp. 1257–1272, 1988.
  • L.B. Almeida, “Backpropagation in perceptrons with feedback,” in Neural Computers, pp. 199–208, Springer, New York, NY, USA, 1988.
  • F. J. Pineda, “Generalization of back-propagation to recurrent neural networks,” Physical Review Letters, vol. 59, no. 19, pp. 2229–2232, 1987.
  • R. Rohwer and B. Forrest, “Training time-dependence in neural networks,” in Proceedings of the 1st IEEE International Conference on Neural Networks, pp. 701–708, San Diego, Calif, USA, 1987.
  • M. Forti and A. Tesi, “New conditions for global stability of neural networks with application to linear and quadratic programming problems,” IEEE Transactions on Circuits and Systems. I, vol. 42, no. 7, pp. 354–366, 1995.
  • Y. Xia and J. Wang, “A general methodology for designing globally convergent optimization neural networks,” IEEE Transactions on Neural Networks, vol. 9, no. 6, pp. 1331–1343, 1998.
  • Y. S. Xia and J. Wang, “On the stability of globally projected dynamical systems,” Journal of Optimization Theory and Applications, vol. 106, no. 1, pp. 129–150, 2000.
  • J.-H. Li, A. N. Michel, and W. Porod, “Analysis and synthesis of a class of neural networks: linear systems operating on a closed hypercube,” IEEE Transactions on Circuits and Systems, vol. 36, no. 11, pp. 1405–1422, 1989.
  • I. Varga, G. Elek, and S. H. Zak, “On the brain-state-in-a-convex-domain neural models,” Neural Networks, vol. 9, no. 7, pp. 1173–1184, 1996.
  • M. A. Cohen and S. Grossberg, “Absolute stability of global pattern formation and parallel memory storage by competitive neural networks,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 13, no. 5, pp. 815–826, 1983.
  • H. Qiao, J. Peng, Z. B. Xu, and B. Zhang, “A reference model approach to stability analysis of neural networks,” IEEE Transactions on Systems, Man, and Cybernetics, Part B, vol. 33, no. 6, pp. 925–936, 2003.
  • X. Yang, X. Liao, Y. Tang, and D. J. Evans, “Guaranteed attractivity of equilibrium points in a class of delayed neural networks,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 16, no. 9, pp. 2737–2743, 2006.
  • H. Zhao, “Global asymptotic stability of Hopfield neural network involving distributed delays,” Neural Networks, vol. 17, no. 1, pp. 47–53, 2004.
  • J. Cao and J. Wang, “Global asymptotic and robust stability of recurrent neural networks with time delays,” IEEE Transactions on Circuits and Systems I, vol. 52, no. 2, pp. 417–426, 2005.
  • Z. Huang, X. Wang, and F. Gao, “The existence and global attractivity of almost periodic sequence solution of discrete-time neural networks,” Physics Letters A, vol. 350, no. 3-4, pp. 182–191, 2006.
  • Z. B. Xu, H. Qiao, J. Peng, and B. Zhang, “A comparative study of two modeling approaches in neural networks,” Neural Networks, vol. 17, no. 1, pp. 73–85, 2004.
  • M. Wang and L. Wang, “Global asymptotic robust stability of static neural network models with S-type distributed delays,” Mathematical and Computer Modelling, vol. 44, no. 1-2, pp. 218–222, 2006.
  • P. Li and J. Cao, “Stability in static delayed neural networks: a nonlinear measure approach,” Neurocomputing, vol. 69, no. 13–15, pp. 1776–1781, 2006.
  • Z. Huang and Y. Xia, “Exponential p-stability of second order Cohen-Grossberg neural networks with transmission delays and learning behavior,” Simulation Modelling Practice and Theory, vol. 15, no. 6, pp. 622–634, 2007.
  • K. Zhao and Y. Li, “Robust stability analysis of fuzzy neural network with delays,” Mathematical Problems in Engineering, vol. 2009, Article ID 826908, 13 pages, 2009.
  • R. Yang, H. Gao, and P. Shi, “Novel robust stability criteria for stochastic Hopfield neural networks with time delays,” IEEE Transactions on Systems, Man, and Cybernetics, Part B, vol. 39, no. 2, pp. 467–474, 2009.
  • R. Yang, Z. Zhang, and P. Shi, “Exponential stability on stochastic neural networks with discrete interval and distributed delays,” IEEE Transactions on Neural Networks, vol. 21, no. 1, pp. 169–175, 2010.
  • X. Li and J. Shen, “LMI approach for stationary oscillation of interval neural networks with discrete and distributed time-varying delays under impulsive perturbations,” IEEE Transactions on Neural Networks, vol. 21, no. 10, pp. 1555–1563, 2010.
  • Y. Zhao, L. Zhang, S. Shen, and H. Gao, “Robust stability criterion for discrete-time uncertain markovian jumping neural networks with defective statistics of modes transitions,” IEEE Transactions on Neural Networks, vol. 22, no. 1, pp. 164–170, 2011.
  • X. W. Li, H. J. Gao, and X. H. Yu, “A Unified Approach to the Stability of Generalized Static Neural Networks With Linear Fractional Uncertainties and Delays,” IEEE Transactions on Systems, Man and Cybernetics, Part B, vol. 41, no. 5, pp. 1275–1286, 2011.
  • H. Dong, Z. Wang, and H. Gao, “Robust ${H}_{\infty }$ filtering for a class of nonlinear networked systems with multiple stochastic communication delays and packet dropouts,” IEEE Transactions on Signal Processing, vol. 58, no. 4, pp. 1957–1966, 2010.
  • W. Zhang, Y. Tang, J.-A. Fang, and X. Wu, “Stability of delayed neural networks with time-varying impulses,” Neural Networks, vol. 36, pp. 59–63, 2012.
  • H. Huang, T. Huang, and X. Chen, “Global exponential estimates of delayed stochastic neural networks with Markovian switching,” Neural Networks, vol. 36, pp. 136–145, 2012.
  • Q. T. Gan, “Exponential synchronization of stochastic Cohen-Grossberg neural networks with mixed time-varying delays and reactio-diffusion via periodically intermittent control,” Neural Networks, vol. 31, pp. 12–21, 2012.
  • Y. K. Li and C. Wang, “Existence and global exponential stability of equilibrium for discretetime fuzzy BAM neural networks with variable delays and impulses,” Fuzzy Sets and Systems, 2012.
  • T. Yang and L.-B. Yang, “The global stability of fuzzy cellular neural network,” IEEE Transactions on Circuits and Systems. I, vol. 43, no. 10, pp. 880–883, 1996.
  • D. Xu and Z. Yang, “Impulsive delay differential inequality and stability of neural networks,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 107–120, 2005.
  • Z. Yang and D. Xu, “Impulsive effects on stability of Cohen-Grossberg neural networks with variable delays,” Applied Mathematics and Computation, vol. 177, no. 1, pp. 63–78, 2006.
  • R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, vol. 2, Cambridge University Press, Cambridge, UK, 1991.
  • Y. Li, “Global exponential stability of BAM neural networks with delays and impulses,” Chaos, Solitons and Fractals, vol. 24, no. 1, pp. 279–285, 2005.