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Journal of Applied Mathematics
Volume 2013, Article ID 396903, 21 pages
http://dx.doi.org/10.1155/2013/396903
Research Article

LMI Approach to Exponential Stability and Almost Sure Exponential Stability for Stochastic Fuzzy Markovian-Jumping Cohen-Grossberg Neural Networks with Nonlinear -Laplace Diffusion

1Institution of Mathematics, Yibin University, Yibin, Sichuan 644007, China
2School of Science Mathematics, University of Electronic Science and Technology of China, Chengdu 610054, China
3College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China

Received 3 February 2013; Accepted 23 March 2013

Academic Editor: Qiankun Song

Copyright © 2013 Ruofeng Rao et al. 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,” IEEE 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 · View at MathSciNet
  2. X. Zhang, S. Wu, and K. Li, “Delay-dependent exponential stability for impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1524–1532, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. X. Wang, R. Rao, and S. Zhong, “LMI approach to stability analysis of Cohen-Grossberg neural networks with p-Laplace diffusion,” Journal of Appplied Mathematics, vol. 2012, Article ID 523812, 12 pages, 2012. View at Publisher · View at Google Scholar
  4. Q. Zhu and X. Li, “Exponential and almost sure exponential stability of stochastic fuzzy delayed Cohen-Grossberg neural networks,” Fuzzy Sets and Systems, vol. 203, pp. 74–94, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. Q. Liu and R. Xu, “Periodic solutions of a Cohen-Grossberg-type BAM neural networks with distributed delays and impulses,” Journal of Applied Mathematics, vol. 2012, Article ID 643418, 17 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. H. Xiang and J. Cao, “Periodic oscillation of fuzzy Cohen-Grossberg neural networks with distributed delay and variable coefficients,” Journal of Applied Mathematics, vol. 2008, Article ID 453627, 18 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. X. Zhou and S. Zhong, “Riccati equations and delay-dependent BIBO stabilization of stochastic systems with mixed delays and nonlinear perturbations,” Advances in Difference Equations, vol. 2010, Article ID 494607, 14 pages, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Y. Zhao and Y. Ma, “Stability of neutral-type descriptor systems with multiple time-varying delays,” Advances in Difference Equations, vol. 2012, 7 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  9. A. Friedman, Stochastic Differential Equations and Applications. Vol. 2, Probability and Mathematical Statistics, vol. 28, Academic Press, New York, NY, USA, 1976. View at MathSciNet
  10. K. Mathiyalagan, R. Sakthivel, and S. Marshal Anthoni, “Exponential stability result for discrete-time stochastic fuzzy uncertain neural networks,” Physics Letters A, vol. 376, no. 8-9, pp. 901–912, 2012. View at Publisher · View at Google Scholar
  11. Q. Ling and H. Deng, “A new proof to the necessity of a second moment stability condition of discrete-time Markov jump linear systems with real states,” Journal of Applied Mathematics, vol. 2012, Article ID 642480, 10 pages, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. Q. Zhu and J. Cao, “Exponential stability of stochastic neural networks with both Markovian jump parameters and mixed time delays,” IEEE Transactions on Systems, Man, and Cybernetics, Part B, vol. 41, no. 2, pp. 341–353, 2011. View at Publisher · View at Google Scholar · View at Scopus
  13. Q. Zhu and J. Cao, “Stability analysis for stochastic neural networks of neutral type with both Markovian jump parameters and mixed time delays,” Neurocomputing, vol. 73, no. 13-15, pp. 2671–2680, 2010. View at Publisher · View at Google Scholar · View at Scopus
  14. Q. Zhu, X. Yang, and H. Wang, “Stochastically asymptotic stability of delayed recurrent neural networks with both Markovian jump parameters and nonlinear disturbances,” Journal of the Franklin Institute, vol. 347, no. 8, pp. 1489–1510, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. Q. Zhu and J. Cao, “Stochastic stability of neural networks with both Markovian jump parameters and continuously distributed delays,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 490515, 20 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. S. Blythe, X. Mao, and X. Liao, “Stability of stochastic delay neural networks,” Journal of the Franklin Institute. Engineering and Applied Mathematics, vol. 338, no. 4, pp. 481–495, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. Buhmann and K. Schulten, “Influence of noise on the function of a “physiological” neural network,” Biological Cybernetics, vol. 56, no. 5-6, pp. 313–327, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. Haykin, Neural Networks, Prentice-Hall, Upper Saddle River, NJ, USA, 1994.
  19. Y. Sun and J. Cao, “pth moment exponential stability of stochastic recurrent neural networks with time-varying delays,” Nonlinear Analysis: Real World Applications, vol. 8, no. 4, pp. 1171–1185, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. L. Wan and J. Sun, “Mean square exponential stability of stochastic delayed Hopfield neural networks,” Physics Letters A, vol. 343, no. 4, pp. 306–318, 2005. View at Publisher · View at Google Scholar · View at Scopus
  21. L. Wan and Q. Zhou, “Convergence analysis of stochastic hybrid bidirectional associative memory neural networks with delays,” Physics Letters A, vol. 370, no. 5-6, pp. 423–432, 2007. View at Publisher · View at Google Scholar · View at Scopus
  22. X. Liang and L. Wang, “Exponential stability for a class of stochastic reaction-diffusion Hopfield neural networks with delays,” Journal of Applied Mathematics, vol. 2012, Article ID 693163, 12 pages, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. Y. Zhang, “Asymptotic stability of impulsive reaction-diffusion cellular neural networks with time-varying delays,” Journal of Applied Mathematics, vol. 2012, Article ID 501891, 17 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. A. Salem, “Invariant regions and global existence of solutions for reaction-diffusion systems with a tridiagonal matrix of diffusion coefficients and nonhomogeneous boundary conditions,” Journal of Applied Mathematics, vol. 2007, Article ID 12375, 15 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. D. J. Higham and T. Sardar, “Existence and stability of fixed points for a discretised nonlinear reaction-diffusion equation with delay,” Applied Numerical Mathematics, vol. 18, no. 1–3, pp. 155–173, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. V. K. Baranwal, R. K. Pandey, M. P. Tripathi, and O. P. Singh, “An analytic algorithm for time fractional nonlinear reaction-diffusion equation based on a new iterative method,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 10, pp. 3906–3921, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. G. Meral and M. Tezer-Sezgin, “The comparison between the DRBEM and DQM solution of nonlinear reaction-diffusion equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 10, pp. 3990–4005, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. F. Liang, “Blow-up and global solutions for nonlinear reaction-diffusion equations with nonlinear boundary condition,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 3993–3999, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  29. T. Yang and L. Yang, “The global stability of fuzzy cellular neural networks,” IEEE Transactions on Circuits and Systems I, vol. 43, no. 10, pp. 880–883, 1996. View at Publisher · View at Google Scholar
  30. Y. Xia, Z. Yang, and M. Han, “Lag synchronization of unknown chaotic delayed yang-yang-type fuzzy neural networks with noise perturbation based on adaptive control and parameter identification,” IEEE Transactions on Neural Networks, vol. 20, no. 7, pp. 1165–1180, 2009. View at Publisher · View at Google Scholar · View at Scopus
  31. Y. Xia, Z. Yang, and M. Han, “Synchronization schemes for coupled identical Yang-Yang type fuzzy cellular neural networks,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 9-10, pp. 3645–3659, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. D. He and D. Xu, “Attracting and invariant sets of fuzzy Cohen-Grossberg neural networks with time-varying delays,” Physics Letters A, vol. 372, no. 47, pp. 7057–7062, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. Y. Liu and W. Tang, “Exponential stability of fuzzy cellular neural networks with constant and time-varying delays,” Physics Letters A, vol. 323, no. 3-4, pp. 224–233, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. S. Niu, H. Jiang, and Z. Teng, “Exponential stability and periodic solutions of FCNNs with variable coefficients and time-varying delays,” Neurocomputing, vol. 71, no. 13-15, pp. 2929–2936, 2008. View at Publisher · View at Google Scholar · View at Scopus
  35. Q. Song and J. Cao, “Impulsive effects on stability of fuzzy Cohen-Grossberg neural networks with time-varying delays,” IEEE Transactions on Systems, Man, and Cybernetics, Part B, vol. 37, no. 3, pp. 733–741, 2007. View at Publisher · View at Google Scholar · View at Scopus
  36. P. Balasubramaniam, V. Vembarasan, and R. Rakkiyappan, “Leakage delays in T-S fuzzy cellular neural networks,” Neural Processing Letters, vol. 33, no. 2, pp. 111–136, 2011. View at Publisher · View at Google Scholar · View at Scopus
  37. K. Mathiyalagan, R. Sakthivel, and S. Marshal Anthoni, “New stability and stabilization criteria for fuzzy neural networks with various activation functions,” Physica Scripta, vol. 84, no. 1, Article ID 015007, 2011. View at Publisher · View at Google Scholar · View at Scopus
  38. S. Muralisankar, N. Gopalakrishnan, and P. Balasubramaniam, “An LMI approach for global robust dissipativity analysis of T-S fuzzy neural networks with interval time-varying delays,” Expert Systems with Applications, vol. 39, no. 3, pp. 3345–3355, 2012. View at Publisher · View at Google Scholar
  39. P. Balasubramaniam, V. Vembarasan, and R. Rakkiyappan, “Delay-dependent robust asymptotic state estimation of Takagi-Sugeno fuzzy Hopfield neural networks with mixed intervaltime-varying delays,” Expert Systems with Applications, vol. 39, no. 1, pp. 472–481, 2012. View at Publisher · View at Google Scholar
  40. P. Balasubramaniam, V. Vembarasan, and R. Rakkiyappan, “Delay-dependent robust exponential state estimation of Markovian jumping fuzzy Hopfield neural networks with mixed random time-varying delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 4, pp. 2109–2129, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. P. Balasubramaniam and V. Vembarasan, “Robust stability of uncertain fuzzy BAM neural networks of neutral-type with Markovian jumping parameters and impulses,” Computers & Mathematics with Applications, vol. 62, no. 4, pp. 1838–1861, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  42. K. Mathiyalagan, R. Sakthivel, and S. M. Anthoni, “New robust passivity criteria for stochastic fuzzy BAM neural networks with time-varying delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 3, pp. 1392–1407, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. R. Sakthivel, A. Arunkumar, K. Mathiyalagan, and S. M. Anthoni, “Robust passivity analysis of fuzzy Cohen-Grossbert BAM neural networks with time-varying delays,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3799–3809, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  44. K. Mathiyalagan, R. Sakthivel, and S. M. Anthoni, “New robust exponential stability results for discrete-time switched fuzzy neural networks with time delays,” Computers & Mathematics with Applications, vol. 64, no. 9, pp. 2926–2938, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  45. R. Sakthivel, K. Mathiyalagan, and S. Marshal Anthoni, “Design of a passification controller for uncertain fuzzy Hopfield neural networks with time-varying delays,” Physica Scripta, vol. 84, no. 4, Article ID 045024, 2011. View at Publisher · View at Google Scholar
  46. P. Vadivel, R. Sakthivel, K. Mathiyalagan, and P. Thangaraj, “Robust stabilisation of non-linear uncertain Takagi-Sugeno fuzzy systems by H control,” Control Theory & Applications, vol. 6, no. 16, pp. 2556–2566, 2012. View at Publisher · View at Google Scholar
  47. C. K. Ahn, “Some new results on stability of Takagi-Sugeno fuzzy Hopfield neural networks,” Fuzzy Sets and Systems, vol. 179, pp. 100–111, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  48. C. K. Ahn, “Switched exponential state estimation of neural networks based on passivity theory,” Nonlinear Dynamics, vol. 67, no. 1, pp. 573–586, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  49. C. K. Ahn, “Exponential stable learning method for Takagi-Sugeno fuzzy delayed neural networks: a convex optimization approach,” Computers & Mathematics with Applications, vol. 63, no. 5, pp. 887–895, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  50. C. K. Ahn, “Delay-dependent state estimation of T-S fuzzy delayed Hopfield neural networks,” Nonlinear Dynamics, vol. 61, no. 3, pp. 483–489, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  51. C. K. Ahn, “Takagi-Sugeno fuzzy Hopfield neural networks for nonlinear system identification,” Neural Processing Letters, vol. 34, no. 1, pp. 59–70, 2011. View at Publisher · View at Google Scholar
  52. C. K. Ahn, “H state estimation for Takagi-Sugeno fuzzy delayed Hopfield neural networks,” International Journal of Computational Intelligence Systems, vol. 4, no. 5, pp. 855–862, 2011. View at Publisher · View at Google Scholar
  53. C. K. Ahn, “Linear matrix inequality optimization approach to exponential robust filtering for switched Hopfield neural networks,” Journal of Optimization Theory and Applications, vol. 154, no. 2, pp. 573–587, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  54. C. K. Ahn, “Exponentially convergent state estimation for delayed switched recurrent neural networks,” The European Physical Journal E, vol. 34, no. 11, p. 122, 2011. View at Google Scholar
  55. C. K. Ahn and M. K. Song, “L2 -L filtering for time-delayed switched hopfield neural networks,” International Journal of Innovative Computing, Information and Control, vol. 7, no. 4, pp. 1831–1843, 2011. View at Google Scholar · View at Scopus
  56. C. K. Ahn, “Passive learning and input-to-state stability of switched Hopfield neural networks with time-delay,” Information Sciences, vol. 180, no. 23, pp. 4582–4594, 2010. View at Publisher · View at Google Scholar · View at Scopus
  57. C. K. Ahn, “An approach to stability analysis of switched Hopfield neural networks with time-delay,” Nonlinear Dynamics, vol. 60, no. 4, pp. 703–711, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  58. J. Pan and S. Zhong, “Dynamic analysis of stochastic reaction-diffusion Cohen-Grossberg neural networks with delays,” Advances in Difference Equations, vol. 2009, Article ID 410823, 18 pages, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  59. R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, vol. 317 of Fundamental Principles of Mathematical Sciences, Springer, Berlin, Germany, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  60. R. F. Rao, S. M. Zhong, and X. R. Wang, “Stochastic stability criteria with LMI conditions for Markovian jumping impulsive BAM neural networkswithmode-dependent time-varying delays and nonlinear reaction-diffusion,” submitted to Communications in Nonlinear Science and Numerical Simulation.
  61. R. A. Adams and J. J. F. Fournier, Sobolev Spaces, vol. 140 of Pure and Applied Mathematics, Elsevier/Academic Press, Amsterdam, The Netherlands, 2nd edition, 2003. View at MathSciNet
  62. L. Wang, Z. Zhang, and Y. Wang, “Stochastic exponential stability of the delayed reaction-diffusion recurrent neural networks with Markovian jumping parameters,” Physics Letters A, vol. 372, no. 18, pp. 3201–3209, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  63. R. Sh. Liptser and A. N. Shiryayev, Theory of martingales, vol. 49 of Mathematics and Its Applications (Soviet Series), Kluwer Academic Publishers, Dordrecht, The Netherlands, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  64. X. Nie and J. Cao, “Stability analysis for the generalized Cohen-Grossberg neural networks with inverse Lipschitz neuron activations,” Computers & Mathematics with Applications, vol. 57, no. 9, pp. 1522–1536, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  65. Y. Y. Wang, L. Xie, and C. E. de Souza, “Robust control of a class of uncertain nonlinear systems,” Systems & Control Letters, vol. 19, no. 2, pp. 139–149, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  66. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, vol. 15 of SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1994. View at Publisher · View at Google Scholar · View at MathSciNet