About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 109319, 14 pages
http://dx.doi.org/10.1155/2012/109319
Research Article

Global Convergence for Cohen-Grossberg Neural Networks with Discontinuous Activation Functions

1School of Mathematics and Physics, Anhui University of Technology, Ma'anshan 243002, China
2School of Computer Science, Anhui University of Technology, Ma'anshan 243002, China

Received 12 September 2012; Accepted 23 October 2012

Academic Editor: Sabri Arik

Copyright © 2012 Yanyan Wang and Jianping Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. A. Cohen and S. Grossberg, “Absolute stability of global pattern formation and parallel memory storage by competitive neural networks,” IIEEE Transactions on Systems, Man, and Cybernetics, vol. 13, no. 5, pp. 815–826, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. K. Gopalsamy, “Global asymptotic stability in a periodic Lotka-Volterra system,” Australian Mathematical Society B, vol. 27, no. 1, pp. 66–72, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. X. Liao, S. Yang, S. Chen, and Y. Fu, “Stability of general neural networks with reaction-diffusion,” Science in China F, vol. 44, pp. 389–395, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. W. Lu and T. Chen, “New conditions on global stability of Cohen-Grossberg neural networks,” Neural Computation, vol. 15, no. 5, pp. 1173–1189, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  5. K. Yuan and J. Cao, “An analysis of global asymptotic stability of delayed Cohen-Grossberg neural networks via nonsmooth analysis,” IEEE Transactions on Circuits and Systems I, vol. 52, no. 9, pp. 1854–1861, 2005. View at Publisher · View at Google Scholar
  6. J. Cao and X. Li, “Stability in delayed Cohen-Grossberg neural networks: LMI optimization approach,” Physica D, vol. 212, no. 1-2, pp. 54–65, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. C. C. Hwang, C. J. Cheng, and T. L. Liao, “Globally exponential stability of generalized Cohen-Grossberg neural networks with delays,” Physics Letters A, vol. 319, no. 1-2, pp. 157–166, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  8. J. Cao and J. Liang, “Boundedness and stability for Cohen-Grossberg neural network with time-varying delays,” Journal of Mathematical Analysis and Applications, vol. 296, no. 2, pp. 665–685, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. J. Zhang, Y. Suda, and H. Komine, “Global exponential stability of Cohen-Grossberg neural networks with variable delays,” Physics Letters A, vol. 338, no. 1, pp. 44–50, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  10. T. Chen and L. Rong, “Robust global exponential stability of Cohen-Grossberg neural networks with time delays,” IEEE Transactions on Neural Networks, vol. 15, no. 1, pp. 203–205, 2004. View at Publisher · View at Google Scholar · View at Scopus
  11. M. Forti and P. Nistri, “Global convergence of neural networks with discontinuous neuron activations,” IEEE Transactions on Circuits and Systems I, vol. 50, no. 11, pp. 1421–1435, 2003. View at Publisher · View at Google Scholar
  12. H. Harrer, J. A. Nossek, and R. Stelzl, “An analog implementation of discrete-time cellular neural networks,” IEEE Transactions on Neural Networks, vol. 3, no. 3, pp. 466–476, 1992. View at Publisher · View at Google Scholar · View at Scopus
  13. M. P. Kennedy and L. O. Chua, “Neural networks for nonlinear programming,” IEEE Transactions on Circuits and Systems I, vol. 35, no. 5, pp. 554–562, 1988. View at Publisher · View at Google Scholar
  14. L. O. Chua, C. A. Desoer, and E. S. Kuh, Linear and Nonlinear Circuits, McGraw-Hill, New York, NY, USA, 1987.
  15. V. I. Utkin, Sliding Modes and Their Application in Variable Structure Systems, U.S.S.R.: MIR, Moscow, Russia, 1978.
  16. J.-P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin, Germany, 1984. View at Publisher · View at Google Scholar
  17. B. E. Paden and S. S. Sastry, “A calculus for computing Filippov's differential inclusion with application to the variable structure control of robot manipulators,” IEEE Transactions on Circuits and Systems, vol. 34, no. 1, pp. 73–82, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. M. Forti, “M-matrices and global convergence of discontinuous neural networks,” International Journal of Circuit Theory and Applications, vol. 35, no. 2, pp. 105–130, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  19. W. Lu and T. Chen, “Dynamical behaviors of delayed neural network systems with discontinuous activation functions,” Neural Computation, vol. 18, no. 3, pp. 683–708, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. W. Lu and T. Chen, “Dynamical behaviors of Cohen-Grossberg neural networks with discontinuous activation functions,” Neural Networks, vol. 18, no. 3, pp. 231–242, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  21. A. F. Filippov, “Differential equations with discontinuous right-hand side,” Transactions of the American Mathematical Society, vol. 42, pp. 199–231, 1964.
  22. M. Hirsch, “Convergent activation dynamics in continuous time networks,” Neural Networks, vol. 2, pp. 331–349, 1989.
  23. D. Hershkowitz, “Recent directions in matrix stability,” Linear Algebra and its Applications, vol. 171, pp. 161–186, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. H. Wu, “Global stability analysis of a general class of discontinuous neural networks with linear growth activation functions,” Information Sciences, vol. 179, no. 19, pp. 3432–3441, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. G. Huang and J. Cao, “Multistability of neural networks with discontinuous activation function,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 10, pp. 2279–2289, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. H. Wu and C. Shan, “Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses,” Applied Mathematical Modelling, vol. 33, no. 6, pp. 2564–2574, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. C. Huang and J. Cao, “Stochastic dynamics of nonautonomous Cohen-Grossberg neural networks,” Abstract and Applied Analysis, vol. 2011, Article ID 297147, 17 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. X. Yang, C. Huang, D. Zhang, and Y. Long, “Dynamics of Cohen-Grossberg neural networks with mixed delays and impulses,” Abstract and Applied Analysis, vol. 2008, Article ID 432341, 14 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. Q. Liu and W. Zheng, “Bifurcation of a Cohen-Grossberg neural network with discrete delays,” Abstract and Applied Analysis, vol. 2012, Article ID 909385, 11 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. M. J. Park, O. M. Kwon, J. H. Park, S. M. Lee, and E. J. Cha, “Synchronization criteria for coupled stochastic neural networks with time-varying delays and leakage delay,” Journal of the Franklin Institute, vol. 349, no. 5, pp. 1699–1720, 2012. View at Publisher · View at Google Scholar
  31. O. M. Kwon, S. M. Lee, J. H. Park, and E. J. Cha, “New approaches on stability criteria for neural networks with interval time-varying delays,” Applied Mathematics and Computation, vol. 218, no. 19, pp. 9953–9964, 2012. View at Publisher · View at Google Scholar
  32. O. M. Kwon, J. H. Park, S. M. Lee, and E. J. Cha, “A new augmented Lyapunov-Krasovskii functional approach to exponential passivity for neural networks with time-varying delays,” Applied Mathematics and Computation, vol. 217, no. 24, pp. 10231–10238, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH