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Exponential input-to-state stability of stochastic Cohen–Grossberg neural networks with mixed delays

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Abstract

In this paper, we study an issue of input-to-state stability analysis for a class of impulsive stochastic Cohen–Grossberg neural networks with mixed delays. The mixed delays consist of varying delays and continuously distributed delays. To the best of our knowledge, the input-to-state stability problem for this class of stochastic system has still not been solved, despite its practical importance. The main aim of this paper is to fill the gap. By constricting several novel Lyapunov–Krasovskii functionals and using some techniques such as the It\(\hat{o}\) formula, Dynkin formula, impulse theory, stochastic analysis theory, and the mathematical induction, we obtain some new sufficient conditions to ensure that the considered system with/without impulse control is mean-square exponentially input-to-state stable. Moreover, the obtained results are illustrated well with two numerical examples and their simulations.

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References

  1. Liao, X., Li, C., Wong, K.: Criteria for exponential stability of Cohen–Grossberg neural networks. Neural Netw. 17, 1401–1414 (2004)

    Article  MATH  Google Scholar 

  2. Guo, S., Huang, L.: Stability analysis of Cohen–Grossberg neural networks. IEEE Trans. Neural Netw. 17, 106–117 (2006)

    Article  Google Scholar 

  3. Wang, Z., Zhang, H.: Global asymptotic stability of reactiondiffusion Cohen–Grossberg neural networks with continuously distributed delays. IEEE Trans. Neural Netw. 21, 39–49 (2010)

    Article  Google Scholar 

  4. Wang, Z., Zhang, H., Yu, W.: Robust stability criteria for interval Cohen–Grossberg neural networks with time varying delay. Neurocomputing 72, 1105–1110 (2009)

    Article  Google Scholar 

  5. Wang, Z., Zhang, H., Li, P.: An LMI approach to stability analysis of reaction-diffusion Cohen–Grossberg neural networks concerning dirichlet boundary conditions and distributed delays. IEEE Trans. Syst. Man Cybern. Part B 40, 1596–1606 (2010)

    Article  Google Scholar 

  6. Song, Q., Cao, J.: Global exponential robust stability of Cohen–Grossberg neural network with time-varying delays and reactiondiffusion terms. J. Franklin Inst. 343, 705–719 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Balasubramaniam, P., Ali, M.S.: Robust exponential stability of uncertain fuzzy Cohen–Grossberg neural networks with time-varying delays. Fuzzy Set. Syst. 161, 608–618 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kao, Y., Wang, C.: Global stability analysis for stochastic coupled reaction diffusion systems on networks. Nonlinear Anal.: RWA 14, 1457–1465 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lu, K., Xu, D., Yang, Z.: Global attraction and stability for Cohen–Grossberg neural networks with delays. Neural Netw. 19, 1538–1549 (2006)

    Article  MATH  Google Scholar 

  10. Song, Q., Zhang, J.: Global exponential stability of impulsive Cohen–Grossberg neural network with time-varying delays. Nonlinear Anal.: RWA 9, 500–510 (2008)

    Article  MATH  Google Scholar 

  11. Oliveira, J.J.: Global stability of a Cohen–Grossberg neural network with both time-varying and continuous distributed delays. Nonlinear Anal.: RWA 12, 2861–2870 (2011)

    Article  MATH  Google Scholar 

  12. Lian, J., Zhang, K.: Exponential stability for switched Cohen–Grossberg neural networks with average dwell time. Nonlinear Dyn. 63, 331–343 (2011)

    Article  MathSciNet  Google Scholar 

  13. Zhang, Z., Zhang, T., Huang, S., Xiao, P.: New global exponential stability result to a general Cohen–Grossberg neural networks with multiple delays. Nonlinear Dyn. 67, 2419–2432 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wu, H., Tao, F., Qin, L., Shi, R., He, L.: Robust exponential stability for interval neural networks with delays and non-Lipschitz activation functions. Nonlinear Dyn. 66, 479–487 (2012)

    Article  MathSciNet  Google Scholar 

  15. Zhao, W.: Global exponential stability analysis of Cohen–Grossberg neural network with delays. Commun. Nonlinear Sci. Numer. Simul. 13, 847–856 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Orman, Z., Arik, S.: New results for global stability of Cohen–Grossberg neural networks with multiple time delays. Neurocomputing 71, 3053–3063 (2008)

    Article  Google Scholar 

  17. Cao, J., Li, X.: Stability in delayed Cohen–Grossberg neural networks: LMI optimization approach. Physica D 212, 54–65 (2005)

  18. Chen, T., Rong, L.: Robust global exponential stability of Cohen–Grossberg neural networks with time delays. IEEE Trans. Neural Netw. 15, 203–206 (2004)

    Article  Google Scholar 

  19. Zhang, H., Wang, Z., Liu, D.: Robust stability analysis for interval Cohen–Grossberg neural networks with unknown time-varying delays. IEEE Trans. Neural Netw. 19, 1942–1955 (2008)

    Article  Google Scholar 

  20. Cichocki, A., Unbehauen, R.: Neural Networks for Optimalition and Signal Processing. Wiley, New York (1993)

    Google Scholar 

  21. Chua, L.O., Yang, L.: Cellular neural networks: applications. IEEE Trans. Circuits Syst. I 35, 1257–1272 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  22. Joya, G., Atencia, M.A., Sandoval, F.: Hopfield neural networks for optimization: study of the different dynamics. Neurocomputing 43, 219–237 (2002)

    Article  MATH  Google Scholar 

  23. Li, W.J., Lee, T.: Hopfield neural networks for affine invariant matching. IEEE Trans. Neural Netw. 12, 1400–1410 (2001)

    Article  Google Scholar 

  24. Wang, S., Fu, D., Xu, M., Hu, D.: Advanced fuzzy cellular neural network: application to CT liver images. Artif. Intell. Med. 39, 65–77 (2007)

    Article  Google Scholar 

  25. Young, S., Scott, P., Nasrabadi, N.: Object recognition using multilayer Hopfield neural network. IEEE Trans. Image Process. 6, 357–372 (1997)

    Article  Google Scholar 

  26. Wang, Z., Liu, Y., Li, M., Liu, X.: Stability analysis for stochastic Cohen–Grossberg neural networks with mixed time delays. IEEE Trans. Neural Netw. 17, 814–820 (2006)

  27. Wang, C., Kao, Y., Yang, G.: Exponential stability of impulsive stochastic fuzzy reaction–diffusion Cohen–Grossberg neural networks with mixed delays. Neurocomputing 89, 55–63 (2012)

    Article  Google Scholar 

  28. Li, T., Song, A., Fei, S.: Robust stability of stochastic Cohen–Grossberg neural networks with mixed time-varying delays. Neurocomputing 73, 542–551 (2009)

    Article  Google Scholar 

  29. Zhu, Q., Li, X.: Exponential and almost sure exponential stability of stochastic fuzzy delayed Cohen–Grossberg neural networks. Fuzzy Set. Syst. 203, 74–94 (2012)

    Article  MATH  Google Scholar 

  30. Fu, X., Li, X.: LMI conditions for stability of impulsive stochastic Cohen–Grossberg neural networks with mixed delays. Commun. Nonlinear Sci. Numer. Simul. 16, 435–454 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Su, W., Chen, Y.: Global robust stability criteria of stochastic Cohen–Grossberg neural networks with discrete and distributed timevarying delays. Commun. Nonlinear Sci. Numer. Simul. 14, 520–528 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Song, Q., Wang, Z.: Stability analysis of impulsive stochastic Cohen–Grossberg neural networks with mixed time delays. Physica A 387, 3314–3326 (2008)

    Article  Google Scholar 

  33. Zhang, H., Wang, Y.: Stability analysis of Markovian jumping stochastic Cohen–Grossberg neural networks with mixed time delays. IEEE Trans. Neural Netw. 19, 366–370 (2008)

    Article  Google Scholar 

  34. Rakkiyappan, R., Balasubramaniam, P.: Dynamic analysis of Markovian jumping impulsive stochastic Cohen–Grossberg neural networks with discrete interval and distributed time-varying delays. Nonlinear Anal. Hybrid Syst. 3, 408–417 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhu, Q., Cao, J.: Robust exponential stability of Markovian jump impulsive stochastic Cohen–Grossberg neural networks with mixed time delays. IEEE Trans. Neural Netw. 21, 1314–1325 (2010)

    Article  Google Scholar 

  36. Zhu, Q., Cao, J.: Mean-square exponential input-to-state stability of stochastic delayed neural networks. Neurocomputing 131, 157–163 (2014)

    Article  Google Scholar 

  37. Zhou, W., Yang, Z.: Input-to-state stability for dynamical neural networks with time-varying delays. Abstr. Appl. Anal. 2012, Article ID 372324, 12 pp (2012)

  38. Yang, Z., Zhou, W., Huang, T.: Exponential input-to-state stability of recurrent neural networks with multiple time-varying delays. Cogn. Neurodyn. 8, 47–54 (2014)

  39. Shi, Y., Zhu, P.: Adaptive synchronization of different Cohen Grossberg chaotic neural networks with unknown parameters and time-varying delays. Nonlinear Dyn. 73, 1721–1728 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zhu, Q., Cao, J.: pth moment exponential synchronization for stochastic delayed Cohen–Grossberg neural networks with Markovian switching. Nonlinear Dyn. 67, 829–845 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The work of the first author was jointly supported by the National Natural Science Foundation of China (61374080), the Natural Science Foundation of Zhejiang Province (LY12F 03010), the Natural Science Foundation of Ningbo (2012A61 0032), and a Project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions; Jinde Cao’s work was jointly supported by the National Natural Science Foundation of China (61272530, 11072059), and the Specialized Research Fund for the Doctoral Program of Higher Education (20110092110017); R. Rakkiyappan’s work was supported by NBHM Research Project.

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Authors and Affiliations

  1. School of Mathematical Sciences, Institute of Finance and Statistics, Nanjing Normal University, Nanjing, 210023, Jiangsu, China

    Quanxin Zhu

  2. Department of Mathematics, Southeast University, Nanjing, 210096, China

    Jinde Cao

  3. Department of Mathematics, Bharathiar University, Coimbatore, 641 046, Tamilnadu, India

    R. Rakkiyappan

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  1. Quanxin Zhu
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  2. Jinde Cao
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  3. R. Rakkiyappan
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Correspondence to Quanxin Zhu.

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Zhu, Q., Cao, J. & Rakkiyappan, R. Exponential input-to-state stability of stochastic Cohen–Grossberg neural networks with mixed delays. Nonlinear Dyn 79, 1085–1098 (2015). https://doi.org/10.1007/s11071-014-1725-2

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  • DOI: https://doi.org/10.1007/s11071-014-1725-2

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