Skip to main content
SpringerLink
Log in

On estimates of solutions to systems of differential equations of neutral type with periodic coefficients

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We study the systems of differential equations of neutral type with periodic coefficients. We establish sufficient conditions for the asymptotic stability of the zero solution and obtain estimates for solutions which characterize the decay rate at infinity.

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. El’sgol’ts L. E. and Norkin S. B., Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press, New York and London (1973).

    MATH  Google Scholar 

  2. Kolmanovskii V. B. and Myshkis A. D., Introduction to the Theory and Applications of Functional-Differential Equations, Kluwer Acad. Publ., Dordrecht (1999) (Math. Appl.; V. 463).

    Book  MATH  Google Scholar 

  3. Gu K., Kharitonov V. L., and Chen J., Stability of Time-Delay Systems. Control Engineering, Birkhäuser, Boston (2003).

    Book  MATH  Google Scholar 

  4. Korenevskiĭ D. G., The Destabilizing Effect of Parametric White Noise in Continuous and Discrete Dynamical Systems [in Russian], Akademperiodika, Kiev (2008).

    Google Scholar 

  5. Halanay A., “Stability theory of linear periodic systems with delay,” Acad. Répub. Popul. Roum., Rev. Math. Pures Appl., 6, No. 4, 633–653 (1961).

    MathSciNet  MATH  Google Scholar 

  6. Hahn W., “On difference differential equations with periodic coefficients,” J. Math. Anal. Appl., 3, No. 1, 70–101 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  7. Stokes A. P., “A Floquet theory for functional differential equations,” Proc. Nat. Acad. Sci. USA, 48, No. 8, 1330–1334 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  8. Zverkin A. M., “Differential difference equations with periodic coefficients,” in: Supplement to the Russian translation of the book by Bellman R. E. and Cooke K. L., Differential-Difference Equations, Mir, Moscow, 1967, pp. 498–535.

    Google Scholar 

  9. Shimanov S. N., Stability of Linear Systems with Periodic Coefficients and Delay [in Russian], Izdat. Ural Univ., Sverdlovsk (1983).

    Google Scholar 

  10. Hale J. K., Theory of Functional Differential Equations, Springer-Verlag, New York, Heidelberg, and Berlin (1977).

    Book  MATH  Google Scholar 

  11. Germanovich O. P., Linear Periodic Equations of Neutral Type and Their Applications [in Russian], Izdat. Leningrad Univ., Leningrad (1986).

    Google Scholar 

  12. Komlenko Yu. V. and Tonkov E. L., “The Lyapunov-Floquet representation for differential equations with aftereffect,” Russian Math. (Iz. VUZ), 39, No. 10, 38–43 (1995).

    MathSciNet  MATH  Google Scholar 

  13. Dolgiĭ Yu. F., Stability of Periodic Differential-Difference Equations [in Russian], Izdat. Ural Univ., Ekaterinburg (1996).

    Google Scholar 

  14. Shil’man S. V., The Method of Generating Functions in the Theory of Dynamical Systems [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  15. Malygina V. V., “On stability of equations with periodic parameters,” in: Functional-Differential Equations [in Russian], Perm’, 1987, pp. 41–43.

    Google Scholar 

  16. Azbelev N. V., Maksimov V. P., and Rakhmatullina L. F., Introduction to the Theory of Functional-Differential Equations [in Russian], Nauka, Moscow (1991).

    Google Scholar 

  17. Berezanskiĭ L. M., “Development of N. V. Azbelev’s W-method in problems of the stability of solutions of linear functional-differential equations,” Differential Equations, 22, No. 5, 521–529 (1986).

    Google Scholar 

  18. Kolmanovskiĭ V. B. and Nosov V. R., Stability and Periodic Regimes of Control Systems with Aftereffect [in Russian], Nauka, Moscow (1981).

    Google Scholar 

  19. Dolgiĭ Yu. F. and Kim A. V., “On the method of Lyapunov functionals for systems with aftereffect,” Differential Equations, 27, No. 8, 918–922 (1991).

    MathSciNet  Google Scholar 

  20. Aleksenko N. V. and Romanovskiĭ R. K., “The method of Lyapunov functionals for linear difference-differential systems with almost periodic coefficients,” Differential Equations, 37, No. 2, 159–165 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  21. Romanovskiĭ R. K. and Trotsenko G. A., “The method of Lyapunov functionals for neutral type linear difference-differential systems with almost periodic coefficients,” Siberian Math. J., 44, No. 2, 355–362 (2003).

    Article  MathSciNet  Google Scholar 

  22. Khusainov D. Ya. and Kozhametov A. T., “Convergence of solutions of nonautonomous systems of neutral type,” Russian Math. (Iz. VUZ), 50, No. 1, 65–69 (2006).

    MathSciNet  MATH  Google Scholar 

  23. Demidenko G. V. and Matveeva I. I., “Asymptotic properties of solutions to delay differential equations,” Vestnik Novosibirsk Univ. Ser. Mat. Mekh. Inform., 5, No. 3, 20–28 (2005).

    MATH  Google Scholar 

  24. Demidenko G. V. and Matveeva I. I., “Stability of solutions to delay differential equations with periodic coefficients of linear terms,” Siberian Math. J., 48, No. 5, 824–836 (2007).

    Article  MathSciNet  Google Scholar 

  25. Demidenko G. V. and Matveeva I. I., “On stability of solutions to linear systems with periodic coefficients,” Siberian Math. J., 42, No. 2, 282–296 (2001).

    Article  MathSciNet  Google Scholar 

  26. Daleckiĭ Ju. L. and Kreĭn M. G., Stability of Solutions to Differential Equations in Banach Space, Amer. Math. Soc., Providence, RI (1974).

    Google Scholar 

  27. Matveeva I. I., “Estimates of solutions to a class of systems of nonlinear delay differential equations,” J. Appl. Indust. Math., 7, No. 4, 557–566 (2013).

    Article  MathSciNet  Google Scholar 

  28. Demidenko G. V., “Stability of solutions to linear differential equations of neutral type,” J. Anal. Appl., 7, No. 3, 119–130 (2009).

    MathSciNet  MATH  Google Scholar 

  29. Demidenko G. V., Vodop’yanov E. S., and Skvortsova M. A., “Estimates of solutions to the linear differential equations of neutral type with several delays of the argument,” J. Appl. Indust. Math., 7, No. 4, 472–479 (2013).

    Article  MathSciNet  Google Scholar 

  30. Kharitonov V., Mondié S., and Collado J., “Exponential estimates for neutral time-delay systems: an LMI approach,” IEEE Trans. Automat. Control, 50, No. 5, 666–670 (2005).

    Article  MathSciNet  Google Scholar 

  31. Baštinec J., Diblík J., Khusainov D.Ya., and Ryvolová A., “Exponential stability and estimation of solutions of linear differential systems of neutral type with constant coefficients,” Bound. Value Probl., 2010, Art. ID 956121, 20 pp.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk, Russia

    G. V. Demidenko & I. I. Matveeva

Authors
  1. G. V. Demidenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. I. Matveeva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. V. Demidenko.

Additional information

Dedicated to Yu. G. Reshetnyak.

Original Russian Text Copyright © 2014 Demidenko G.V. and Matveeva I.I.

The authors were supported by the Russian Foundation for Basic Research (Grant 13-01-00329) and the Siberian Division of the Russian Academy of Sciences (Grant No. 80).

__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 5, pp. 1059–1077, September–October, 2014.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Demidenko, G.V., Matveeva, I.I. On estimates of solutions to systems of differential equations of neutral type with periodic coefficients. Sib Math J 55, 866–881 (2014). https://doi.org/10.1134/S0037446614050061

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446614050061

Keywords

Search

Navigation