Skip to main content
SpringerLink
Log in

Delay differential equations under nonlinear impulsive control and applications to neural network models

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

In this paper, a class of delay differential equations with nonlinear impulsive control is discussed. Based on the nonsmooth analysis, criteria of stability are obtained for delay differential equations with nonlinear impulses control under certain conditions. These criteria can be applied to some neural network models. At the end of the paper, two examples are provided to illustrate the feasibility and effectiveness of the proposed 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

Similar content being viewed by others

References

  1. T. Erneux, Applied Delay Differential Equations, Springer, Berlin, 2009.

    MATH  Google Scholar 

  2. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Volume 191 in the series of Mathematics in Science and Engineering, Academic Press, New York, 1993.

    MATH  Google Scholar 

  3. J. Wiener and J. K. Hale, Ordinary and Delay Differential Equations, J. Wiley, New York, 1992.

    Google Scholar 

  4. K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population, Kluwer Academic Press, Dordrecht, 1992.

    MATH  Google Scholar 

  5. J. Kuang and Y. H. Cong, Stability of Numerical Methods for Delay Differential Equations, Science Press, Beijing, 2005.

    Google Scholar 

  6. M. Forti, On global asymptotic stability of a class of nonlinear system arising in neural networks theory, J. Diff. Equs., 1994, 113: 246–264.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. D. Cao and L. Wang, Periodic oscillatory solution of bidirectional associative memory networks with delays, Phys. Rev., 2000, 61: 1825–1828.

    Article  MathSciNet  Google Scholar 

  8. S. J. Guo and L. H. Huang, Periodic oscillation for a class of neural networks with variable coefficients, Nonlin. Anal., 2004, 6: 545–561.

    MathSciNet  Google Scholar 

  9. H. T. Lu, Global exponential stability analysis of Cohen-Grossberg neural networks, IEEE Trans. Circuits & Systems II: Express Briefs, 2005, 52: 476–479.

    Article  Google Scholar 

  10. H. Akca, R. Alassar, and V. Corachav, Continuous-time additive Hopfield-type networks with impulses, J. Math. Anal. Appl., 2004, 290: 436–451.

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Mohamad and K. Gopalsamy, Exponential stability preservation in semi-discretisations of networks with nonlinear impulses, Commun. Nonlin. Sci. Numer. Simul., 2009, 14: 27–50.

    Article  MathSciNet  MATH  Google Scholar 

  12. P. van den Driessche and X. Zou, Global atractivity in delayed Hopfield neural networks models, SIAM. J. Appl. Math., 1998, 58: 1878–1890.

    Article  MathSciNet  MATH  Google Scholar 

  13. Z. K. Huang and Y. H. Xia, Exponential p-stability of second order Cohen-Grossberg neural networks with transmission delays and learning behavior, Simul. Modelling Prac. Theory, 2007, 15: 622–634.

    Article  Google Scholar 

  14. Y. Zhao, Q. S. Lu, and Z. Feng, Stability for the mix-delayed Cohen-Grossberg neural networks with nonlinear impulse, Journal of Systems Science & Complexity, 2010, 23(3): 665–680.

    Article  MathSciNet  MATH  Google Scholar 

  15. K. Yuan and J. D. Cao, An analysis of global asymptotic stability of delayed Cohen-Grossberg neural networks via nonsmooth analysis, IEEE Trans. Circuits Syst., 2005, 345: 1854–1861.

    MathSciNet  Google Scholar 

  16. Z. Wen and J. T. Sun, Global Asymptotic stability of delayed BAM neural networks with impulses via nonsmooth analysis, Neurocomputing, 2008, 71: 1543–1549.

    Article  Google Scholar 

  17. H. Guan, D. J. Hill and X. M. Shen, On hybrid impulsive and switching systems and application to nonlinear control, IEEE Trans. Auto. Control, 2005, 50: 1058–1062

    Article  MathSciNet  Google Scholar 

  18. C. X. Li and J. T. Sun, Stability analysis of nonlinear stochastic differential delay systems under impulsive control, Phys. Lett., 2010, 374: 1154–1158.

    Article  MathSciNet  Google Scholar 

  19. S. Mohamad and K. Gopalsamy, A unified treatment for stability preservation in computer simulations of impulsive BAM networks, Comput. Math. Appl., 2008, 55: 2043–2063.

    Article  MathSciNet  MATH  Google Scholar 

  20. Z. K. Huang and Y. H. Xia, Global exponential stability of BAM neural networks with transmission delays and nonlinear impulsive, Chaos, Solitons and Fractals, 2008, 38: 489–498.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. T. Rockafellar and R. J. B. Wets, Variational Analysis, Springer-Verlag, Berlin/Heideberg, 1998.

    Book  MATH  Google Scholar 

  22. F. H. Clarke, Optimization a Nonsmooth Analysis, Wiley, New York, 1983.

    Google Scholar 

  23. B. H. Pourciau, Hadamard theorem for Lacally Lipschitzian maps, J. Math. Anal. Appl., 1982, 85: 279–285.

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Science, Academic Press, New York, 1994.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Dynamics and Control, Beihang University, Beijing, 100191, China

    Yong Zhao & Qishao Lu

  2. Department of Mathematics, University of Texas-Pan American, Edinburg, TX, 78539, USA

    Zhaosheng Feng

  3. Department of Mathematics, Zhejiang Normal University, Jinhua, 210034, China

    Yonghui Xia

Authors
  1. Yong Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qishao Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhaosheng Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yonghui Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhaosheng Feng.

Additional information

This paper is supported by Natural Science Foundation of China under Grant Nos. 10972018 and 11072013.

This paper was recommended for publication by Editor Bingyu ZHANG.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhao, Y., Lu, Q., Feng, Z. et al. Delay differential equations under nonlinear impulsive control and applications to neural network models. J Syst Sci Complex 25, 707–719 (2012). https://doi.org/10.1007/s11424-012-1110-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-012-1110-5

Key words

Search

Navigation