Skip to main content
SpringerLink
Log in

Robust Stability for Interval Stochastic Neural Networks with Time-Varying Discrete and Distributed Delays

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

This paper investigates the problem of robust stability for a class of stochastic interval neural networks with discrete and distributed time-varying delays. The discrete delays are assumed to be varying within a given interval, while the parameter uncertainties are assumed to be bounded in some given compact sets. Based on the Itô differential formula and stochastic stability theory, delay-range-dependent criteria for stochastic interval neural networks with time-varying delays are derived to guarantee robust global asymptotic stability in mean square. In this paper, when estimating the upper bound of the derivative of Lyapunov functionals, we are concerned with better handling of terms related to the discrete and distributed delays and establishing less conservative results. These robust stability conditions are presented in terms of linear matrix inequalities and can be efficiently solved via standard numerical software. An important feature of the results proposed in this paper is that the stability conditions are dependent on the upper and lower bounds of the discrete delays. Numerical examples are given to illustrate the effectiveness of the proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Blythe S., Mao X., Liao X.: Stability of stochastic delay neural networks. J. Franklin Inst. 338(4), 481–495 (2004)

    Article  MathSciNet  Google Scholar 

  2. Cao J., Wang J.: Global asymptotic stability of a general class of recurrent neural networks with time-varying delays. IEEE Trans. Circuits Syst. Fund. Theory Appl. 50(1), 34–44 (2003)

    Article  MathSciNet  Google Scholar 

  3. Cao J., Liang J., Lam J.: Exponential stability of high-order bidirectional associative memory neural networks with time delays. Physica D 199(3–4), 425–436 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen W., Lu X.: Mean square exponential stability of uncertain stochastic delayed neural networks. Phys. Lett. A 372(7), 1061–1069 (2008)

    Article  MathSciNet  Google Scholar 

  5. Chen W., Zheng W.: Global asymptotic stability of a class of neural networks with distributed delays. IEEE Trans. Circuits Syst. I Regul. Pap. 53(3), 644–652 (2006)

    Article  MathSciNet  Google Scholar 

  6. Feng J., Xu S., Zou Y.: Delay-dependent stability of neutral type neural networks with distributed delays. Neurocomputing 72(10–12), 2576–2580 (2009)

    Article  Google Scholar 

  7. Gao H., Chen T., Lam J.: A new delay system approach to network-based control. Automatica 44(1), 39–52 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gu K., Kharitonov V., Chen J.: Stability of Time-Delay Systems. Birkhauser, Boston (2003)

    Book  MATH  Google Scholar 

  9. Guan Z., Lam J., Chen G.: On impulsive autoassociative neural networks. Neural Netw. 13(1), 63–69 (2000)

    Article  Google Scholar 

  10. Han Q.L., Gu K.: Stability of linear systems with time-varying delay: a generalized discretized Lyapunov functional approach. Asian J. Control 3(3), 170–180 (2001)

    Article  Google Scholar 

  11. He Y., Wu M., She J., Liu G.: Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties. IEEE Trans Automat Control 49(5), 828–832 (2004)

    Article  MathSciNet  Google Scholar 

  12. He Y., Liu G., Rees D.: New delay-dependent stability criteria for neural networks with time-varying delay. IEEE Trans. Neural Netw. 18(1), 310 (2007)

    Article  MathSciNet  Google Scholar 

  13. He Y., Wang Q., Lin C., Wu M.: Delay-range-dependent stability for systems with time-varying delay. Automatica 43(2), 371–376 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ho D.W.C., Liang J., Lam J.: Global exponential stability of impulsive high-order BAM neural networks with time-varying delays. Neural Netw. 19(10), 1581–1590 (2006)

    Article  MATH  Google Scholar 

  15. Huang C., Cao J.: Almost sure exponential stability of stochastic cellular neural networks with unbounded distributed delays. Neurocomputing 72(13–15), 3352–3356 (2009)

    Article  MATH  Google Scholar 

  16. Huang H., Feng G.: Delay-dependent stability for uncertain stochastic neural networks with time-varying delay. Physica A 381, 93–103 (2007)

    Article  Google Scholar 

  17. Lam J., Gao H., Wang C.: Stability analysis for continuous systems with two additive time-varying delay components. Syst. Control Lett. 56(1), 16–24 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li C., Liao X., Zhang R.: Global robust asymptotical stability of multi-delayed interval neural networks: an LMI approach. Phys. Lett. A 328(6), 452–462 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li C., Chen J., Huang T.: A new criterion for global robust stability of interval neural networks with discrete time delays. Chaos Solitons Fractals 31(3), 561–570 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li H., Chen B., Zhou Q., Fang S.: Robust exponential stability for uncertain stochastic neural networks with discrete and distributed time-varying delays. Phys. Lett. A 372(19), 3385–3394 (2008)

    Article  MathSciNet  Google Scholar 

  21. Li T., Song A., Fei S.: Robust stability of stochastic Cohen–Grossberg neural networks with mixed time-varying delays. Neurocomputing 73(1–3), 542–551 (2009)

    Article  Google Scholar 

  22. Lin C., Wang Q., Lee T.: A less conservative robust stability test for linear uncertain time-delay systems. IEEE Trans. Automat. Control 51(1), 87–91 (2006)

    Article  MathSciNet  Google Scholar 

  23. Lin C., Wang Z., Yang F.: Observer-based networked control for continuous-time systems with random sensor delays. Automatica 45(2), 578–584 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu Y., Wang Z., Liu X.: Global exponential stability of generalized recurrent neural networks with discrete and distributed delays. Neural Netw. 19(5), 667–675 (2006)

    Article  MATH  Google Scholar 

  25. Liu Y., Wang Z., Liang J., Liu X.: Stability and synchronization of discrete-time Markovian jumping neural networks with mixed mode-dependent time delays. IEEE Trans. Neural Netw. 20(7), 1102–1116 (2009)

    Article  Google Scholar 

  26. Mou S., Gao H., Lam J., Qiang W.: A new criterion of delay-dependent asymptotic stability for Hopfield neural networks with time delay. IEEE Trans. Neural Netw. 19(3), 532–535 (2008)

    Article  Google Scholar 

  27. Shao H.: New delay-dependent stability criteria for systems with interval delay. Automatica 45(3), 744–749 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Shu Z., Lam J.: Global exponential estimates of stochastic interval neural networks with discrete and distributed delays. Neurocomputing 71(13–15), 2950–2963 (2008)

    Article  Google Scholar 

  29. Shu Z., Lam J.: Exponential estimates and stabilization of uncertain singular systems with discrete and distributed delays. Int. J. Control 81(6), 865–882 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Shustin E., Fridman E.: On delay-derivative-dependent stability of systems with fast-varying delays. Automatica 43(9), 1649–1655 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Song Q., Cao J.: Global robust stability of interval neural networks with multiple time-varying delays. Math. Comput. Simul. 74(1), 38–46 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Song Q., Cao J.: Robust stability in Cohen–Grossberg neural network with both time-varying and distributed delays. Neural Process. Lett. 27(2), 179–196 (2008)

    Article  MATH  Google Scholar 

  33. Su W., Chen Y.: Global robust exponential stability analysis for stochastic interval neural networks with time-varying delays. Commun. Nonlinear Sci. Numer. Simulat. 14(5), 2293–2300 (2009)

    Article  MathSciNet  Google Scholar 

  34. Tang Y., Fang J., Xia M., Yu D.: Delay-distribution-dependent stability of stochastic discrete-time neural networks with randomly mixed time-varying delays. Neurocomputing 72(16–18), 3830–3838 (2009)

    Article  Google Scholar 

  35. Wang Y., Xie L., De Souza C.: Robust control of a class of uncertain nonlinear systems. Syst. Control Lett. 19(2), 139–149 (1992)

    Article  MathSciNet  Google Scholar 

  36. 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(3), 814–820 (2006)

    Article  Google Scholar 

  37. Wang Z., Shu H., Liu Y., Ho D.W.C., Liu X.: Robust stability analysis of generalized neural networks with discrete and distributed time delays. Chaos Solitons Fractals 30(4), 886–896 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wang Z., Fang J., Liu X.: Global stability of stochastic high-order neural networks with discrete and distributed delays. Chaos Solitons Fractals 36(2), 388–396 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wang Z., Ho D.W.C., Liu Y., Liu X.: Robust H ∞ control for a class of nonlinear discrete time-delay stochastic systems with missing measurements. Automatica 45(3), 684–691 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wang Z., Wang Y., Liu Y.: Global synchronization for discrete-time stochastic complex networks with randomly occurred nonlinearities and mixed time-delays. IEEE Trans. Neural Netw. 21(1), 11–25 (2010)

    Article  Google Scholar 

  41. Wu Z., Su H., Chu J., Zhou W.: Improved result on stability analysis of discrete stochastic neural networks with time delay. Phys. Lett. A 373(17), 1546–1552 (2009)

    Article  MathSciNet  Google Scholar 

  42. Xu S., Lam J.: A new approach to exponential stability analysis of neural networks with time-varying delays. Neural Netw. 19(1), 76–83 (2006)

    Article  MATH  Google Scholar 

  43. Xu S., Lam J.: A survey of linear matrix inequality techniques in stability analysis of delay systems. Int. J. Syst. Sci. 39(12), 1095–1113 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  44. Xu S., Lam J., Ho D.W.C.: Novel global robust stability criteria for interval neural networks with multiple time-varying delays. Phys. Lett. A 342(4), 322–330 (2005)

    Article  MATH  Google Scholar 

  45. Xu S., Lam J., Ho D.: Delay-dependent asymptotic stability of neural networks with time-varying delays. Int. J. Bifurcat. Chaos 18(1), 245–250 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  46. Yang R., Gao H., Lam J., Shi P.: New stability criteria for neural networks with distributed and probabilistic delays. Circuits Syst. Signal Process. 28(4), 505–522 (2009)

    MathSciNet  MATH  Google Scholar 

  47. Yue D., Tian E., Zhang Y., Peng C.: Delay-distribution-dependent stability and stabilization of T-S fuzzy systems with probabilistic interval delay. IEEE Trans. Syst. Man Cybern. B Cybern. 39(2), 503–516 (2009)

    Article  Google Scholar 

  48. Zhang Y., Tian E.: Novel robust selay-dependent exponential stability criteria for stochastic delayed recurrent neural networks. Int. J. Innov. Comput. Inform. Control 5(9), 2735–2744 (2009)

    Google Scholar 

  49. Zhang J., Shi P., Qiu J.: Novel robust stability criteria for uncertain stochastic Hopfield neural networks with time-varying delays. Nonlinear Anal. Real World Appl. 8(4), 1349–1357 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  50. Zhang B., Xu S., Zou Y.: Improved delay-dependent exponential stability criteria for discrete-time recurrent neural networks with time-varying delays. Neurocomputing 72(1–3), 321–330 (2008)

    Article  Google Scholar 

  51. Zhang Z., Mou S., Lam J., Gao H.: New passivity criteria for neural networks with time-varying delay. Neural Netw. 22(7), 864–868 (2009)

    Article  Google Scholar 

  52. Zhou Q., Chen B., Lin C., Li H.: Mean square exponential stability for uncertain delayed stochastic neural networks with Markovian jump parameters. Circuits Syst. Signal Process. 29(2), 331–348 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Hongyi Li & Huijun Gao

  2. Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, Hong Kong

    K. C. Cheung & James Lam

Authors
  1. Hongyi Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. C. Cheung
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. James Lam
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Huijun Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hongyi Li.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, H., Cheung, K.C., Lam, J. et al. Robust Stability for Interval Stochastic Neural Networks with Time-Varying Discrete and Distributed Delays. Differ Equ Dyn Syst 19, 97–118 (2011). https://doi.org/10.1007/s12591-010-0075-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-010-0075-x

Keywords

Search

Navigation