Skip to main content
SpringerLink
Log in

W 1,2(Ω)- and X 1,2(Ω)-stability of reaction-diffusion cellular neural networks with delay

  • Published:
Science in China Series F: Information Sciences Aims and scope Submit manuscript

Abstract

With Poincare’s inequality and auxiliary function applied in a class of retarded cellular neural networks with reaction-diffusion, the conditions of the systems’ W 1,2(Ω)-exponential and X 1,2(Ω)-asmptotic stability are obtained. The stability conditions containing diffusion term are different from those obtained in the previous papers in their exponential stability conditions. One example is given to illustrate the feasibility of this method in the end.

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. Lu H T, Shen R M, Chung F L. Absolute exponential stability of a class of recurrent neural networks with multiple and variable delays. Theor Comp Sci, 2005, 344(2–3): 103–119

    Article  MATH  MathSciNet  Google Scholar 

  2. Yuan K, Cao J D. An analysis of global asymptotic stability of delayed Cohen-Grossberg neural networks via nonsmooth analysis. IEEE Trans Circ Syst-I: Regular Papers, 2005, 52(9): 1854–1861

    Article  MathSciNet  Google Scholar 

  3. Tu F H, Liao X F, Zhang W. Delay-dependent asymptotic stability of a two-neural system with different time delays. Chaos Solit Fract, 2006, 28(2): 437–447

    Article  MATH  MathSciNet  Google Scholar 

  4. Sabri A, Vedat T. Global asymptotic stability analysis of bidirectional associative memory neural networks with constant time delays. Neurocomputing, 2005, 68(10): 161–176

    Google Scholar 

  5. Liao X X, Wang J, Zeng Z G. Global asymptotic stability and global exponential stability of delayed cellular neural networks. IEEE Trans Circ Syst-II: Express Briefs, 2005, 52(7): 403–409

    Article  Google Scholar 

  6. Zhang Q, Wei X P, Xu J. An improved result for complete stability of delayed cellular neural networks. Automatica, 2005, 41(2): 333–337

    Article  MATH  MathSciNet  Google Scholar 

  7. Cao J D, Daniel W C H. A general framework for global asymptotic stability analysis of delayed neural networks based on LMI approach. Chaos Solit Fract, 2005, 24(5): 1317–1329

    Article  MATH  Google Scholar 

  8. Zhang Y. Global exponential convergence of recurrent neural networks with variable delays. Theor Comp Sci, 2004, 312 (2–3): 281–293

    MATH  Google Scholar 

  9. Lu L, Zhang Z Y, Yang Y R. Global exponential stability in delayed Hopfield neural network models (in Chinese). Acta Electr Sin, 2002, 30(10): 1431–1434

    Google Scholar 

  10. Li X M, Huang L H, Zhu H Y. Global stability of cellular neural networks with constant and variable delays. Nonlinear Anal, Theor, Meth Appl, 2003, 53(3–4): 319–333

    Article  MATH  MathSciNet  Google Scholar 

  11. Liu M Q. Discrete-time delayed standard neural network model and its application. Sci China Ser F-Inf Sci, 2006, 49(2): 137–154

    Article  MATH  Google Scholar 

  12. Liao X X, Yang S Z, Chen S J, et al. Stability of general neural network with reaction-diffusion. Sci China Ser F-Inf Sci, 2001, 44(5): 389–395

    Google Scholar 

  13. Wang L S, Xu D Y. Global exponential stability of Hopfield reaction-diffusion neural networks with time-varying delays. Sci China Ser F-Inf Sci, 2003, 46(6): 466–474

    Article  MATH  Google Scholar 

  14. Liang J L, Cao J D. Global exponential stability of reaction-diffusion recurrent neural networks with time-varying delays. Phys Lett A, 2004, 314(5–6): 434–442

    Article  Google Scholar 

  15. Son Q K, Zhao Z J, Li Y M. Global exponential stability of BAM neural networks with distributed delays and reaction-diffusion terms. Phys Lett A, 2005, 335(2–3): 213–225

    Google Scholar 

  16. Dai Z J, Sun J H. Convergence dynamic of stochastic reaction-diffusion Hopfield neural networks with continuously distributed (in Chinese). J Nanjing Univ (Nat Sci), 2005, 22(2): 197–211

    MATH  MathSciNet  Google Scholar 

  17. Ding N, Zhao H Y. The stability of neural network with reaction-diffusion and distributed delays (in Chinese). J Xingjiang Normal Univ (Nat Sci Ed), 2005, 24(3): 9–11

    Google Scholar 

  18. Zheng W F, Zhang J Y. Globally asymptotic stability of neural networks with reaction-diffusion (in Chinese). J Southwest Jiaotong Univ, 2004, 39(1): 117–120

    MATH  Google Scholar 

  19. Zhao Z J, Song Q K. Exponential periodicity and stability of neural networks with reaction-diffusion terms and both variable and unbounded delays. Comp Math Appl, 2006, 51(3–4): 475–486

    Article  MATH  MathSciNet  Google Scholar 

  20. Qiu J L. Exponential stability of impulsive neural networks with time-varying delays and reaction-diffusion terms. Neurocomputing, 2007, 70(4–6): 1102–1108

    Google Scholar 

  21. Zhou Q H, Wan L, Sun J H. Exponential stability of reaction-diffusion generalized Cohen-Grossberg neural networks with time-varying delays. Chaos Solit Fract, 2007, 32(5): 1713–1719

    Article  MATH  MathSciNet  Google Scholar 

  22. Cui B T, Lou X Y. Global asymptotic stability of BAM neural networks with distributed delays and reaction-diffusion terms. Chaos Solit Fract, 2006, 27(5): 1347–1354

    Article  MATH  MathSciNet  Google Scholar 

  23. Song Q K, Cao J D. Global exponential robust stability of Cohen-Grossberg neural network with time-varying delays and reaction-diffusion terms. J Franklin Institute, 2006, 343(7): 705–719

    Article  MATH  MathSciNet  Google Scholar 

  24. Wang L S, Gao Y Y. Global exponential robust stability of reaction-diffusion interval neural networks with time-varying delays. Phys Lett A, 2006, 350(5–6): 342–348

    Article  MathSciNet  Google Scholar 

  25. Song Q K, Cao J D. Global exponential stability and existence of periodic solutions in BAM networks with delays and reaction-diffusion terms. Chaos Solit Fract, 2005, 23(2): 421–430

    Article  MATH  MathSciNet  Google Scholar 

  26. Lou X Y, Cui T, Wu W. On global exponential stability and existence of periodic solutions for BAM neural networks with distributed delays and reaction-diffusion terms. Chaos Solit Fract, 2008, 36(4): 1044–1054

    Article  MATH  MathSciNet  Google Scholar 

  27. Song Q K, Cao J D, Zhao Z J. Periodic solutions and its exponential stability of reaction-diffusion recurrent neural networks with continuously distributed delays. Nonlinear Anal: Real World Appl, 2006, 7(1): 65–80

    Article  MATH  MathSciNet  Google Scholar 

  28. Allegretto W, Papini D. Stability for delayed reaction-diffusion neural networks. Phys Lett A, 2007, 360(6): 669–680

    Article  MathSciNet  Google Scholar 

  29. Sanchez E N, Perez J P. Input-to-state stability analysis for dynamic NN. IEEE Trans Circuits Syst-I, 1999, 46(11): 1395–1398

    Article  MATH  MathSciNet  Google Scholar 

  30. Liu R Q, Xie S L. The Stability and Control of Distributed Parameter System with Delay (in Chinese). Guozhou: South-China University of Technology Publishers, 1998. 12–14

    Google Scholar 

  31. McOwen R C. Partial Differential Equations: Methods and Applications. Upper Saddle River, NJ: Prentice Hall, Inc, 1996. 163–164

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Hunan Institute of Engineering, Xiangtan, 411101, China

    YiPing Luo, WenHua Xia & GuoRong Liu

  2. Institute of System Engineering, South China University of Technology, Guangzhou, 510640, China

    YiPing Luo & FeiQi Deng

Authors
  1. YiPing Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. WenHua Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. GuoRong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. FeiQi Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to YiPing Luo.

Additional information

Supported by the National Natural Science Foundation of China (Grant No. 60374023), the Natural Science Foundation of Hunan Province (Grant No. 07JJ6112), and Scientific Research Fund of Hunan Provincial Education Department (Grant Nos. 04A012 and 07A015), and the Construct Program of the Key Discipline in Hunan Province (Control Theory and Control Engineering)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Luo, Y., Xia, W., Liu, G. et al. W 1,2(Ω)- and X 1,2(Ω)-stability of reaction-diffusion cellular neural networks with delay. Sci. China Ser. F-Inf. Sci. 51, 1980–1991 (2008). https://doi.org/10.1007/s11432-008-0139-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11432-008-0139-5

Keywords

Search

Navigation