Skip to main content
SpringerLink
Log in

Robust Stability of Markovian Jump Stochastic Neural Networks with Time Delays in the Leakage Terms

  • Published:
Neural Processing Letters Aims and scope Submit manuscript

Abstract

This paper deals with the problem of exponential stability for a class of Markovian jump stochastic neural networks with time delays in the leakage terms and mixed time delays. The jumping parameters are modeled as a continuous-time, finite-state Markov chain, and the mixed time delays consist of time-varying delays and distributed delays. By using the method of model transformation, Lyapunov stability theory, stochastic analysis and linear matrix inequalities techniques, several novel sufficient conditions are derived to guarantee the exponential stability in the mean square of the equilibrium point of the suggested system in two cases: with known or unknown parameters. Moreover, some remarks and discussions are given to illustrate that the obtained results are significant, which comprises and generalizes those obtained in the previous literature. In particular, the obtained stability conditions are delay-dependent, which depends on all the delay constants, and thus the presented results are less conservatism. Finally, two numerical examples are provided to show the effectiveness of the theoretical results.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Arik S (2000) Stability analysis of delayed neural networks. IEEE Trans Circuits Syst I 47(7):1089– 1092

    Article  MATH  MathSciNet  Google Scholar 

  2. Arik S, Orman Z (2005) Global stability analysis of Cohen–Grossberg neural networks with time varying delays. Phys Lett A 341(5–6):410–421

    Article  MATH  Google Scholar 

  3. Joy MP (2000) Results concerning the absolute stability of delayed neural networks. Neural Netw 13(6):613–616

    Article  Google Scholar 

  4. Blythe S, Mao X, Liao X (2001) Stability of stochastic delay neural networks. J Frankl Inst 338(4):481–495

    Article  MATH  MathSciNet  Google Scholar 

  5. Arik S, Tavsanoglu V (2005) Global asymptotic stability analysis of bidirectional associative memory neural networks with constant time delays. Neurocomputing 68:161–176

    Article  Google Scholar 

  6. Huang H, Ho DWC, Lam J (2005) Stochastic stability analysis of fuzzy Hopfield neural networks with time-varying delays. IEEE Trans Circuits Syst II 52(5):251–255

    Article  Google Scholar 

  7. Javidmanesh E, Afsharnezhad Z, Effati S (2013) Existence and stability analysis of bifurcating periodic solutions in a delayed five-neuron BAM neural network model. Nonlinear Dyn 72:149–164

    Article  MATH  MathSciNet  Google Scholar 

  8. Rakkiyappan R, Balasubramaniam P (2008) Delay-dependent asymptotic stability for stochastic delayed recurrent neural networks with time varying delays. Appl Math Comput 198(2):526–533

    Article  MATH  MathSciNet  Google Scholar 

  9. Zhou Q, Wan L (2008) Exponential stability of stochastic delayed Hopfield neural networks. Appl Math Comput 199(1):84–89

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  11. Moon YS, Park P, Kwon WH, Lee YS (2001) Delay-dependent robust stabilization of uncertain state-delayed systems. Int J Control 74(14):1447–1455

    Article  MATH  MathSciNet  Google Scholar 

  12. Chen Y (2002) Global stability of neural networks with distributed delays. Neural Netw 15(7):867–871

    Article  Google Scholar 

  13. Chen W, Zheng W (2007) Delay-dependent robust stabilization for uncertain neutral systems with distributed delays. Automatica 43(1):95–104

    Article  MATH  MathSciNet  Google Scholar 

  14. Park JH (2008) On global stability criterion of neural networks with continuously distributed delays. Chaos Solitons Fractals 37(2):444–449

    Article  MATH  MathSciNet  Google Scholar 

  15. Zhao H (2004) Global asymptotic stability of Hopfield neural network involving distributed delays. Neural Netw 17(1):47–53

    Article  MATH  Google Scholar 

  16. Yang H, Chu T (2007) LMI conditions for stability of neural networks with distributed delays. Chaos Solitons Fractals 34(2):557–563

    Article  MATH  MathSciNet  Google Scholar 

  17. Liu X, Wang Z, Liu X (2006) On global exponential stability of generalized stochastic neural networks with mixed time delays. Neurocomputing 70(1–3):314–326

    Article  Google Scholar 

  18. Wang Z, Liu Y, Liu X (2005) On global asymptotic stability of neural networks with discrete and distributed delays. Phys Lett A 345(4–5):299–308

    Article  MATH  Google Scholar 

  19. Gopalsamy K (2007) Leakage delays in BAM. J Math Anal Appl 325(2):1117–1132

    Article  MATH  MathSciNet  Google Scholar 

  20. Balasubramaniam P, Kalpana M, Rakkiyappan R (2011) Global asymptotic stability of BAM fuzzy cellular neural networks with time delay in the leakage term, discrete and unbounded distributed delays. Math Comput Model 53(5–6):839–853

    Article  MATH  MathSciNet  Google Scholar 

  21. Balasubramaniam P, Kalpana M, Rakkiyappan R (2011) Existence and global asymptotic stability of fuzzy cellular neural networks with time delay in the leakage term and unbounded distributed delays. Circuits Syst Signal Process 30(6):1595–1616

    Article  MATH  MathSciNet  Google Scholar 

  22. Li X, Fu X, Balasubramaniam P, Rakkiyappan R (2010) Existence, uniqueness and stability analysis of recurrent neural networks with time delay in the leakage term under impulsive perturbations. Nonlinear Anal RWA 11(5):4092C4108

    MathSciNet  Google Scholar 

  23. Li X, Rakkiyappan R, Balasubramaniam P (2011) Existence and global stability analysis of equilibrium of fuzzy cellular neural networks with time delay in the leakage term under impulsive perturbations. J Frankl Inst 348(2):135–155

    Article  MATH  MathSciNet  Google Scholar 

  24. Liu B (2013) Global exponential stability for BAM neural networks with time-varying delays in the leakage terms. Nonlinear Anal RWA 14(1):559–566

    Article  MATH  Google Scholar 

  25. Long S, Song Q, Wang X, Li D (2012) Stability analysis of fuzzy cellular neural networks with time delay in the leakage term and impulsive perturbations. J Frankl Inst 349(7):2461–2479

    Article  MATH  MathSciNet  Google Scholar 

  26. Rakkiyappan R, Chandrasekar A, Lakshmanan S, Park JH, Jung HY (2013) Effects of leakage time-varying delays in Markovian jump neural networks with impulse control. Neurocomputing 121(1):365–378

    Article  Google Scholar 

  27. Lakshmanan S, Park JH, Lee TH, Jung HY, Rakkiyappan R (2013) Stability criteria for BAM neural networks with leakage delays and probabilistic time-varying delays. Appl Math Comput 219(17):9408–9423

    Article  MATH  MathSciNet  Google Scholar 

  28. Lakshmanan S, Park JH, Jung HY (2013) Robust delay-dependent stability criteria for dynamic systems with nonlinear perturbations and leakage delay. Circuits Syst Signal Process 32(4):1637–1657

    Article  MathSciNet  Google Scholar 

  29. Balasubramaniam P, Vembarasan V, Rakkiyappan R (2012) Global robust asymptotic stability analysis of uncertain switched Hopfield neural networks with time delay in the leakage term. Neural Comput Appl 21(7):1593–1616

    Article  Google Scholar 

  30. Balasubramaniam P, Vembarasan V, Rakkiyappan R (2011) Leakage delays in T–S fuzzy cellular neural networks. Neural Process Lett 33(2):111–136

    Article  Google Scholar 

  31. Balasubramaniam P, Vembarasan V (2011) Asymptotic stability of BAM neural networks of neutral-type with impulsive effects and time delay in the leakage term. Int J Comput Math 88(15):3271–3291

    Article  MATH  MathSciNet  Google Scholar 

  32. Liu Y, Wang Z, Liu X (2008) On delay-dependent robust exponential stability of stochastic neural networks with mixed time delays and Markovian switching. Nonlinear Dyn 54(3):199–212

    Article  MATH  Google Scholar 

  33. Balasubramaniam P, Vembarasan V (2011) Robust stability of uncertain fuzzy BAM neural networks of neutral-type with Markovian jumping parameters and impulses. Comput Math Appl 62(4):1838–1861

    Article  MATH  MathSciNet  Google Scholar 

  34. Balasubramaniam P, Vembarasan V, Rakkiyappan R (2011) Delay-dependent robust exponential state estimation of Markovian jumping fuzzy Hopfield neural networks with mixed random time-varying delays. Commun Nonlinear Sci Numer Simul 16(4):2109–2129

    Article  MATH  MathSciNet  Google Scholar 

  35. Balasubramaniam P, Rakkiyappan R (2009) Delay-dependent robust stability analysis for Markovian jumping stochastic Cohen–Grossberg neural networks with discrete interval and distributed time-varying delays. Nonlinear Anal Hybrid Syst 3(3):207–214

    Article  MATH  MathSciNet  Google Scholar 

  36. Zhang H, Wang Y (2008) Stability analysis of Markovian jumping stochastic Cohen–Grossberg neural networks with mixed time delays. IEEE Trans Neural Netw 19(2):366–370

    Article  Google Scholar 

  37. Zhu Q, Cao J (2011) Exponential stability of stochastic neural networks with both Markovian jump parameters and mixed time delays. IEEE Trans Syst Man Cybern B 41(2):341–353

    Google Scholar 

  38. Zhu Q, Cao J (2010) Robust exponential stability of Markovian jump impulsive stochastic Cohen–Grossberg neural networks with mixed time delays. IEEE Trans Neural Netw 21(8):1314–1325

    Article  Google Scholar 

  39. Zhu Q, Cao J (2012) Stability analysis of Markovian jump stochastic BAM neural networks with impulse control and mixed time delays. IEEE Trans Neural Netw Learn Syst 23(3):467–479

    Article  Google Scholar 

  40. Gu K (2000) An integral inequality in the stability problem of time-delay systems. In: Proceedings of 39th IEEE conference on decision and control, Sydney, pp 2805–2810

  41. Wang Z, Liu Y, Yu L, Liu X (2006) Exponential stability of delayed recurrent neural networks with Markovian jumping parameters. Phys Lett A 356(4–5):346–352

    Article  MATH  Google Scholar 

  42. Zhang Y (2013) Stochastic stability of discrete-time Markovian jump delay neural networks with impulses and incomplete information on transition probability. Neural Netw 46:276–282

    Article  MATH  Google Scholar 

  43. Zhu S, Shen Y (2013) Robustness analysis for connection weight matrices of global exponential stability of stochastic recurrent neural networks. Neural Netw 38:17–22

    Article  Google Scholar 

  44. Faydasicok O, Arik S (2012) Robust stability analysis of a class of neural networks with discrete time delays. Neural Netw 29–30:52–59

    Article  Google Scholar 

  45. Zhou T, Wang M, Long M (2012) Existence and exponential stability of multiple periodic solutions for a multidirectional associative memory neural network. Neural Process Lett 35(2):187–202

    Article  Google Scholar 

  46. Zheng C, Shan Q, Wang Z (2012) Improved stability results for stochastic Cohen–Grossberg neural networks with discrete and distributed delays. Neural Process Lett 35(2):103–129

    Article  Google Scholar 

  47. Boyd S, Ghaoui L, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. SIAM, Philadelphia

    Book  MATH  Google Scholar 

Download references

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China (61374080, 61272530, 11072059), the Natural Science Foundation of Zhejiang Province (LY12F03010), the Natural Science Foundation of Ningbo (2012A610032), the Natural Science Foundation of Jiangsu Province (BK2012741), the Specialized Research Fund for the Doctoral Program of Higher Education (20110092110017,20130092110017), the Deanship of Scientific Research (DSR), King Abdulaziz University (KAU), under Grant 3-130/1434/HiCi.

Author information

Authors and Affiliations

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

    Quanxin Zhu

  2. Department of Mathematics and Research Center for Complex Systems and Network Sciences, Southeast University, Nanjing , 210096, Jiangsu, China

    Jinde Cao

  3. Department of Mathematics, Quaid-I-Azam University, Islamabad , 44000, Pakistan

    Tasawar Hayat

  4. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

    Jinde Cao & Tasawar Hayat

  5. Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia

    Fuad Alsaadi

Authors
  1. Quanxin Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinde Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tasawar Hayat
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Fuad Alsaadi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Quanxin Zhu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhu, Q., Cao, J., Hayat, T. et al. Robust Stability of Markovian Jump Stochastic Neural Networks with Time Delays in the Leakage Terms. Neural Process Lett 41, 1–27 (2015). https://doi.org/10.1007/s11063-013-9331-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11063-013-9331-8

Keywords

Search

Navigation