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Lasalle method and general decay stability of stochastic neural networks with mixed delays

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Abstract

This paper investigates the general decay pathwise stability conditions on a class of stochastic neural networks with mixed delays by applying Lasalle method. The mixed time delays comprise both time-varying delays and infinite distributed delays. The contributions are as follows: (1) we extend the Lasalle-type theorem to cover stochastic differential equations with mixed delays; (2) based on the stochastic Lasalle theorem and the M-matrix theory, new criteria of general decay stability, which includes the almost surely exponential stability and the almost surely polynomial stability and the partial stability, for neural networks with mixed delays are established. As an application of our results, this paper also considers a two-dimensional delayed stochastic neural networks model.

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References

  1. Kosko, B.: Adaptive bidirectional associative memories. Appl. Opt. 2b, 4947–4960 (1987)

    Article  Google Scholar 

  2. Chua, L.O., Yang, L.: Cellular neural networks: theory and applications. IEEE Trans. Circuits Syst. I 35, 1257–1290 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cohen, M.A., Grossberg, S.: Absolute stability and global pattern formation and parallel memory storage by competitive neural networks. IEEE Trans. Syst. Man Cybern. 13, 815–826 (1983)

    MATH  MathSciNet  Google Scholar 

  4. Shi, P., Boukas, E.K., Shi, Y., Agarwal, R.K.: Optimal guaranteed cost control of uncertain discrete time-delay systems. J. Comput. Appl. Math. 157, 435–451 (2008)

    Article  MathSciNet  Google Scholar 

  5. Song, Q., Wang, Z.: An analysis on existence and global exponential stability of periodic solutions for BAM neural networks with time-varying delays. Nonlinear Anal., Real World Appl. 8, 1224–1234 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Wang, Z., Ho, D.H.C., Liu, X.: A note on the robust stability of uncertain fuzzy systems with time-delays. IEEE Trans. Syst. Man Cybern., Part A 34, 570–576 (2004)

    Article  Google Scholar 

  7. Wang, Z., Shu, H., Fang, J., Liu, X.: Robust stability for stochastic Hopfield neural networks with time-delays. Nonlinear Anal., Real World Appl. 7, 1119–1128 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Wang, F., Wu, H.: Mean square exponential stability and periodic solutions of stochastic interval neural networks with mixed time delays. Neurocomputing 73, 3256–3263 (2010)

    Article  Google Scholar 

  9. Zhou, W., Lu, H., Duan, C.: Exponential stability of hybrid stochastic neural networks with mixed time delays and nonlinearity. Neurocomputing 72, 3357–3365 (2009)

    Article  Google Scholar 

  10. Liu, Y., Wang, Z., Liu, X.: Global exponential stability of generalized recurrent neural networks with discrete and distributed delays. Neural Netw. 19, 667–675 (2006)

    Article  MATH  Google Scholar 

  11. Liu, Yu., Wang, Z., Liu, X.: On global exponential stability of generalized stochastic neural networks with mixed time-delay. Neurocomputing 70, 314–326 (2006)

    Article  Google Scholar 

  12. Liu, Z., Lv, S., Zhong, S.: pth moment exponential synchronization analysis for a class of stochastic neural networks with mixed delays. Commun. Nonlinear Sci. Numer. Simul. 15, 1899–1909 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Haykin, S.: Neural Networks. Prentice-Hall, Englewood Cliffs (1994)

    MATH  Google Scholar 

  14. Balasubramani, P., Rakkiyappan, R.: Global asymptotic stability of stochastic recurrent neural networks with multiple discrete delays and unbounded distributed delays. Appl. Math. Comput. 254, 680–686 (2008)

    Article  Google Scholar 

  15. Hu, S., Liao, X.X., Mao, X.: Stochastic Hopfield neural networks. J. Phys. A, Math. Gen. 36, 2235–2249 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Blythe, S., Mao, X., Shah, A.: Razumikhin-type theorems on stability of stochastic neural networks with delays. Stoch. Anal. Appl. 19, 85–101 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Liu, X., Chen, T.: Robust μ-stability for uncertain stochastic neural networks with unbounded time-varying delays. Physica A 387, 2952–2962 (2008)

    MathSciNet  Google Scholar 

  18. Liao, X.X., Mao, X.: Exponential stability and instability of stochastic neural networks. Stoch. Anal. Appl. 14(2), 165–185 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  19. Liao, X.X., Mao, X.: Stability of stochastic neural networks. Neural Parallel Sci. Comput. 4(2), 205–244 (1996)

    MATH  MathSciNet  Google Scholar 

  20. LaSalle, J.P.: Stability theory of ordinary differential equations. J. Differ. Equ. 4, 57–65 (1968) 19

    Article  MATH  MathSciNet  Google Scholar 

  21. Mao, X.: Stochastic versions of the Lasalle theorem. J. Differ. Equ. 153, 175–195 (1999)

    Article  MATH  Google Scholar 

  22. Mao, X.: Lasalle-type theorems for stochastic differential delay equations. J. Math. Anal. Appl. 236, 350–369 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  23. Mao, X.: A note on the Lasalle-type theorem for stochastic differential delay equations. J. Math. Anal. Appl. 268, 125–142 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  24. Mao, X.: The Lasalle-type theorem for stochastic functional differential equations. Nonlinear Stud. 7, 307–328 (2000)

    MATH  MathSciNet  Google Scholar 

  25. Shen, Y., Luo, Q., Mao, X.: The improved Lasalle-type theorems for stochastic functional differential equations. J. Math. Anal. Appl. 318, 134–154 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  26. Mao, X.: Stochastic Differential Equations and Applications. Horwood, Chichester (1997)

    MATH  Google Scholar 

  27. Hu, Y., Wu, F., Huang, C.: Stochastic Lotka–Volterra models with multiple delays. J. Math. Anal. Appl. 375, 42–57 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  28. Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia (1994)

    Book  MATH  Google Scholar 

  29. Wu, F., Hu, S., Huang, C.: Robustness of general decay stability of nonlinear neutral stochastic functional differential equations with infinite delay. Syst. Control Lett. 59, 195–202 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  30. Shen, Y., Hui, J., Liao, X.: Lasalle-type theorems for neutral stochastic functional differential equations. Control Theory Appl. 23, 221–224 (2006)

    MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, P.R. China

    Yangzi Hu & Chengming Huang

  2. Division of Mathematics, Graduate School of Science, Kyoto University, Kyoto, 606-8502, Japan

    Yangzi Hu

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  1. Yangzi Hu
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  2. Chengming Huang
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Correspondence to Yangzi Hu.

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Hu, Y., Huang, C. Lasalle method and general decay stability of stochastic neural networks with mixed delays. J. Appl. Math. Comput. 38, 257–278 (2012). https://doi.org/10.1007/s12190-011-0477-0

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  • DOI: https://doi.org/10.1007/s12190-011-0477-0

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