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Estimates for Solutions to a Class of Nonautonomous Systems of Neutral Type with Unbounded Delay

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Abstract

Under consideration is the class of nonlinear systems of nonautonomous differential equations of neutral type with a variable delay that can be unbounded. Using a Lyapunov–Krasovskii functional, we establish some estimates of solutions that allow us to conclude whether the solutions are stable. In the case of exponential and asymptotic stability, we estimate the attraction domains and the rate of stabilization of solutions at infinity.

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References

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

    MATH  Google Scholar 

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

    Book  Google Scholar 

  3. Korenevskii D. G., Stability of Dynamical Systems under Random Perturbations of Parameters. Algebraic Criteria, Naukova Dumka, Kiev (1989) [Russian].

    Google Scholar 

  4. Azbelev N. V., Selected Works, Moscow and Izhevsk, Institute of Computer Science (2012) [Russian].

    Google Scholar 

  5. Dolgii Yu. F., Stability of Periodic Differential-Difference Equations, Ural Univ., Yekaterinburg (1996) [Russian].

    Google Scholar 

  6. Khusainov D. Ya. and Shatyrko A. V., The Method of Lyapunov Functions in the Study of the Stability of Functional-Differential Systems, Kiev Univ., Kiev (1997) [Russian].

    MATH  Google Scholar 

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

    Book  Google Scholar 

  8. Michiels W. and Niculescu S. I., Stability, Control, and Computation for Time-Delay Systems. An Eigenvalue-Based Approach, Soc. Indust. Appl. Math., Philadelphia (2014) (Adv. Design Control; Vol. 27).

    Book  Google Scholar 

  9. Agarwal R. P., Berezansky L., Braverman E., and Domoshnitsky A., Nonoscillation Theory of Functional Differential Equations with Applications, Springer, New York (2012).

    Book  Google Scholar 

  10. Kharitonov V. L., Time-Delay Systems. Lyapunov Functionals and Matrices, Birkhäuser, Basel (2013).

    Book  Google Scholar 

  11. Gil’ M. I., Stability of Neutral Functional Differential Equations, Atlantis, Paris (2014) (Atlantis Stud. Diff. Equ.; Vol. 3).

    Book  Google Scholar 

  12. Andreev A. S., “The Lyapunov functionals method in stability problems for functional differential equations,” Automation and Remote Control, vol. 70, no. 9, 1438–1486 (2009).

    Article  MathSciNet  Google Scholar 

  13. Fridman E., “Tutorial on Lyapunov-based methods for time-delay systems,” European J. Control, vol. 20, 271–283 (2014).

    MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Demidenko G. V. and Matveeva I. I., “On estimates of solutions to systems of differential equations of neutral type with periodic coefficients,” Sib. Math. J., vol. 55, no. 5, 866–881 (2014).

    Article  MathSciNet  Google Scholar 

  17. Demidenko G. V. and Matveeva I. I., “Estimates for solutions to a class of nonlinear time-delay systems of neutral type,” Electron. J. Diff. Equ., vol. 2015, no. 34, 1–14 (2015).

    MathSciNet  MATH  Google Scholar 

  18. Demidenko G. V. and Matveeva I. I., “Estimates for solutions to a class of time-delay systems of neutral type with periodic coefficients and several delays,” Electron. J. Qual. Theory Differ. Equ., vol. 2015, no. 83, 1–22 (2015).

    Article  MathSciNet  Google Scholar 

  19. Demidenko G. V. and Matveeva I. I., “Exponential stability of solutions to nonlinear time-delay systems of neutral type,” Electron. J. Diff. Equ., vol. 19, 1–20 (2016).

    MathSciNet  MATH  Google Scholar 

  20. Demidenko G. V., Matveeva I. I., and Skvortsova M. A., “Estimates for solutions to neutral differential equations with periodic coefficients of linear terms,” Sib. Math. J., vol. 60, no. 5, 828–841 (2019).

    Article  MathSciNet  Google Scholar 

  21. Matveeva I. I., “On exponential stability of solutions to periodic neutral-type systems,” Sib. Math. J., vol. 58, no. 2, 264–270 (2017).

    Article  MathSciNet  Google Scholar 

  22. Matveeva I. I., “On the exponential stability of solutions of periodic systems of the neutral type with several delays,” Differ. Equ., vol. 53, no. 6, 725–735 (2017).

    Article  MathSciNet  Google Scholar 

  23. Matveeva I. I., “On the exponential stability of solutions to linear periodic systems of neutral type with variable delay,” Sib. Electr. Math. Reports, vol. 16, 748–756 (2019).

    MATH  Google Scholar 

  24. Matveeva I. I., “Estimates of the exponential decay of solutions to linear systems of neutral type with periodic coefficients,” J. Appl. Ind. Math., vol. 13, no. 3, 511–518 (2019).

    Article  MathSciNet  Google Scholar 

  25. Matveeva I. I., “Exponential stability of solutions to nonlinear time-varying delay systems of neutral type equations with periodic coefficients,” Electron. J. Differ. Equ., vol. 2020, no. 20, 1–12 (2020).

    MathSciNet  Google Scholar 

  26. Matveeva I. I., “Estimates of exponential decay of solutions to one class of nonlinear systems of neutral type with periodic coefficients,” Comp. Math. Math. Phys., vol. 60, no. 4, 601–609 (2020).

    Article  MathSciNet  Google Scholar 

  27. Hartman Ph., Ordinary Differential Equations, Wiley, New York (1964).

    MATH  Google Scholar 

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Acknowledgment

The author is grateful to Professor G. V. Demidenko for useful discussions.

Funding

The author was supported by the Russian Foundation for Basic Research (Grant 18–29–10086).

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Correspondence to I. I. Matveeva.

Additional information

Translated from Sibirskii Matematicheskii Zhurnal, 2021, Vol. 62, No. 3, pp. 583–598. https://doi.org/10.33048/smzh.2021.62.310

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Matveeva, I.I. Estimates for Solutions to a Class of Nonautonomous Systems of Neutral Type with Unbounded Delay. Sib Math J 62, 468–481 (2021). https://doi.org/10.1134/S0037446621030101

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  • DOI: https://doi.org/10.1134/S0037446621030101

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