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Global robust asymptotic stability analysis of uncertain switched Hopfield neural networks with time delay in the leakage term

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Abstract

This paper deals with the problem of delay-dependent global robust asymptotic stability of uncertain switched Hopfield neural networks (USHNNs) with discrete interval and distributed time-varying delays and time delay in the leakage term. Some Lyapunov––Krasovskii functionals are constructed and the linear matrix inequality (LMI) approach are employed to derive some delay-dependent global robust stability criteria which guarantee the global robust asymptotic stability of the equilibrium point for all admissible parametric uncertainties. The proposed results that do not require the boundedness, differentiability, and monotonicity of the activation functions. Moreover, the stability behavior of USHNNs is very sensitive to the time delay in the leakage term. It can be easily checked via the LMI control toolbox in Matlab. In the absence of leakage delay, the results obtained are also new results. Finally, nine numerical examples are given to show the effectiveness of the proposed results.

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References

  1. Haykin S (1998) Neural networks: a comprehensive foundation. Prentice-Hall, New Jersey

    Google Scholar 

  2. Cichoki A, Unbehauen R (1993) Neural networks for optimization and signal processing. Wiley, Chichester

    Google Scholar 

  3. Hopfield JJ (1984) Neurons with graded response have collective computational properties like those of two-state neurons. Proc Nat Acad Sci 81:3088–3092

    Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. Cohen MA, Grossberg S (1983) Absolute stability of global pattern formation and parallel memory storage by competitive neural networks. IEEE Trans Syst Man Cybernet 13:815–825

    MathSciNet  MATH  Google Scholar 

  6. Kosko B (1988) Bi-directional associative memories. IEEE Trans Syst Man Cybernet 18:49–60

    Article  MathSciNet  Google Scholar 

  7. Cao J, Wang J (2005) Global exponential stability and periodicity of recurrent neural networks with time delays. IEEE Trans Circuits Syst I 52:920–931

    Article  MathSciNet  Google Scholar 

  8. Cao J, Ho DWC (2005) A general framework for global asymptotic stability analysis of delayed neural networks based on LMI approach. Chaos Solitons Fractals 24:1317–1329

    Article  MathSciNet  MATH  Google Scholar 

  9. Tian J, Zhou X (2010) Improved asymptotic stability criteria for neural networks with interval time-varying delay. Expert Syst Appl 37:7521–7525

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  11. Cao J, Yuan K, Li H (2006) Global asymptotical stability of recurrent neural networks with multiple discrete delays and distributed delays. IEEE Trans Neural Netw 17:1646–1651

    Article  Google Scholar 

  12. He Y, Liu G, Rees D, Wu M (2007) Stability analysis for neural networks with time-varying interval delay. IEEE Trans Neural Netw 18:1850–1854

    Article  Google Scholar 

  13. Kosko B (1992) Neural networks and fuzzy systems. Prentice Hall, New Delhi

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Gopalsamy K (1992) Stability and oscillations in delay differential equations of population dynamics. Kluwer Academic Publishers, Dordrecht

    MATH  Google Scholar 

  16. Li C, Huang T (2009) On the stability of nonlinear systems with leakage delay. J Franklin Inst 346:366–377

    Article  MathSciNet  MATH  Google Scholar 

  17. 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 Real World Appl 11:4092–4108

    Article  MathSciNet  MATH  Google Scholar 

  18. Li X, Cao J (2010) Delay-dependent stability of neural networks of neutral type with time delay in the leakage term. Nonlinearity 23:1709–1726

    Article  MathSciNet  MATH  Google Scholar 

  19. Park JH (2007) An analysis of global robust stability of uncertain cellular neural networks with discrete and distributed delays. Chaos Solitons Fractals 32:800–807

    Article  MathSciNet  MATH  Google Scholar 

  20. Gao M, Cui B (2009) Global robust stability of neural networks with multiple discrete delays and distributed delays. Chaos Solitons Fractals 40:1823–1834

    Article  MathSciNet  MATH  Google Scholar 

  21. Park JH (2007) An analysis of global robust stability of uncertain cellular neural networks with discrete and distributed delays. Chaos Solitons Fractals 32:800–807

    Article  MathSciNet  MATH  Google Scholar 

  22. Cao J, Chen T (2004) Globally exponentially robust stability and periodicity of delayed neural networks. Chaos Solitons Fractals 22:957–963

    Article  MathSciNet  MATH  Google Scholar 

  23. Ozcan N, Arik S (2006) Global robust stability analysis of neural networks with multiple time delays. IEEE Trans Circuits Syst I 53:166–176

    Article  MathSciNet  Google Scholar 

  24. Sun J, Liu GP, Chen J, Rees D (2009) Improved stability criteria for neural networks with time-varying delay. Phys Lett A 373:342–348

    Article  MATH  Google Scholar 

  25. Farra NH, Mhaskar P, Christofides PD (2005) Output feedback control of switched nonlinear systems using multiple Lyapunov functions. Syst Cont Lett 54:1163–1182

    Article  MATH  Google Scholar 

  26. Liberzon D (2003) Switching in systems and control. In: Systems and control: foundations and applications. Birkhäuser, Boston

  27. Zong GD, Xu SY, Wu YQ (2008) Robust \(H_{\infty}\) stabilization for uncertain switched impulsive control systems with state delay: an LMI approach. Nonlinear Anal Hybrid Syst 2:1287–1300

    Article  MathSciNet  MATH  Google Scholar 

  28. Hespanha JP, Morse AS (2002) Switching between stabilizing controllers. Automatica 38:1905–1917

    Article  MathSciNet  MATH  Google Scholar 

  29. Niamsup P (2009) Stability of time-varying switched systems with time-varying delay. Nonlinear Anal Hybrid Syst 3:631–639

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhang Y, Liu XZ, Shen XM (2007) Stability of switched systems with time delay. Nonlinear Anal Hybrid Syst 1:44–58

    Article  MathSciNet  MATH  Google Scholar 

  31. Tsividis Y (1989) Switched neural networks. United States Patent, Patent number 487,36,61

  32. Brown TX (1989) Neural networks for switching. IEEE Commun Mag 72–81

  33. Muselli M (2003) Gene selection through switched neural networks. Contribution to NETTAB 03, Bologna 27–28

  34. Huang H, Qu YZ, Li HX (2005) Robust stability analysis of switched Hopfield neural networks with time-varying delay under uncertainty. Phys Lett A 345:345–354

    Article  MATH  Google Scholar 

  35. Lou X, Cui B (2007) Delay-dependent criteria for robust stability of uncertain switched Hopfield neural networks. Internat J Automat Comput 4:304–314

    Article  Google Scholar 

  36. Ahn CK (2010) Passive Learning and Input-to-state stability of switched Hopfield neural networks with time-delay. Inf Sci 180:4582–4594

    Article  MATH  Google Scholar 

  37. Ahn CK (2010) An error passivation approach to filtering for switched neural networks with noise disturbance. Neural Comput Appl. doi:10.1007/s00521-010-0474-5

  38. Ahn CK (2010) An \(H_{\infty}\) approach to stability analysis of switched Hopfield neural networks with time-delay. Nonlinear Dyn 60:703–711

    Article  MATH  Google Scholar 

  39. Ahn CK, Song MK (2011) \(L_2-L_\infty\) Filtering for time-delayed switched Hopfield neural networks. Int J Innovative Comput Inform Cont 7:1831–1844

    Google Scholar 

  40. Ahn CK (2011) Switched exponential state estimation of neural networks based on passivity theory. Nonlinear Dyn doi:10.1007/s11071-011-0010-x

  41. Gu K, Kharitonov VL, Chen J (2003) Stability of time-delay systems. Birkhäuser, Boston

    Book  MATH  Google Scholar 

  42. Xu SY, Chen TW (2002) Robust \(H_{\infty}\) control for uncertain stochastic systems with state delay. IEEE Trans Automat Cont 47:2089–2094

    Article  Google Scholar 

  43. Boyd S, Ghaoui LE, Feron E, Balakrishnan V (1994) Linear matrix inequalities in systems and control theory. SIAM, Philadelphia

    Book  Google Scholar 

  44. Sun J, Liu GP, Chen J (2009) Delay-dependent stability and stabilization of neutral time-delay systems. Internat J Robust Nonlinear Cont 19:1364–1375

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors sincerely thank the Associate Editor and anonymous reviewer for their constructive comments and fruitful suggestions to improve the quality of the manuscript. The work of the authors was supported by UGC-SAP (DRS-II) grant no. F.510/2/DRS/2009(SAP-I).

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Authors and Affiliations

  1. Department of Mathematics, Gandhigram Rural University, Gandhigram, Tamilnadu, India

    P. Balasubramaniam & V. Vembarasan

  2. Department of Mathematics, Bharathiyar University, Coimbatore, Tamilnadu, India

    R. Rakkiyappan

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  1. P. Balasubramaniam
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  2. V. Vembarasan
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  3. R. Rakkiyappan
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Correspondence to P. Balasubramaniam.

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Balasubramaniam, P., Vembarasan, V. & Rakkiyappan, R. Global robust asymptotic stability analysis of uncertain switched Hopfield neural networks with time delay in the leakage term. Neural Comput & Applic 21, 1593–1616 (2012). https://doi.org/10.1007/s00521-011-0639-x

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  • DOI: https://doi.org/10.1007/s00521-011-0639-x

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