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On the Choice of Methods for Stabilizing Systems Containing a Linear Delay

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Abstract

We consider the stabilization problem for systems of linear differential equations containing delay that is linearly dependent on the argument (time). Stabilization methods for certain systems with constant coefficients in the right-hand side are proposed. Due to the fact that the delay increases indefinitely in time, the stabilization is considered over an infinite time interval. The numerical examples illustrating the effectiveness of the stabilization methods are presented.

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REFERENCES

  1. B. G. Grebenshchikov and V. I. Rozhkov, “Asymptotic behavior of the solution of a stationary system with delay,” Differ. Equations 29, 640–647 (1993).

    MathSciNet  MATH  Google Scholar 

  2. V. I. Rozhkov and A. M. Popov, “Estimates of the solutions of certain systmes of differential equations with time lag,” Differ. Uravn. 7, 271–278 (1971).

    MATH  Google Scholar 

  3. J. Carr and J. Dyson, “The matrix functional-differential equation,” Proc. R. Soc. Edinburgh, Sect. A, 74, 165–174 (1976).

  4. V. V. Malygina, “On the asymptotic behavior of the solution of one equation with delay,” Vestn. Permsk. Gos. Tekh. Uninv. Prikl. Mat. Mekh., No. 3, 26–29 (1996).

  5. J. R. Ockendon and A. Tayler, “The dynamics of a current collection system for an electric locomotive,” Proc. R. Soc. London Ser. A 322, 447–468 (1971). https://doi.org/10.1098/rspa.1971.0078

    Article  Google Scholar 

  6. B. G. Grebenshchikov, “Construction of almost periodic solutions to a linear delay system,” Sib. Math. J. 57, 788–795 (2016). https://doi.org/10.1134/S0037446616050074

    Article  MathSciNet  MATH  Google Scholar 

  7. N. N. Krasovskii, Theory of Motion Control (Nauka, Moscow, 1968).

    MATH  Google Scholar 

  8. V. D. Furasov, Motion Stability, Estimations, and Stabilization (Nauka, Moscow, 1977).

    Google Scholar 

  9. V. I. Zubov, Lectures on Control Theory (Nauka, Moscow, 1975).

    Google Scholar 

  10. B. G. Grebenshchikov, “On the stabilization of stationary linear systems with delay linearly depending on time,” Preprint No. 4384-92 (VINITI, 1991).

    Google Scholar 

  11. V. D. Furasov, Stability and Stabilization of Discrete Processes (Nauka, Moscow, 1982).

    MATH  Google Scholar 

  12. B. G. Grebenshchikov, “Stability of stationary systems of linear differential equations of neutral type with delay that depends linearly on time,” Soviet Math. 35 (7), 65–67 (1991).

    MathSciNet  MATH  Google Scholar 

  13. A. P. Zhabko, O. G. Tikhomirov, and O. N. Chizhova, “On stabilization of a class of systems with time proportional delay,” Vestn. St. Petersburg Univ. Appl. Math. Comput. Sci. Control Processes 14, 165–172 (2018). https://doi.org/10.21638/11702/spbu10.2018.209

    Article  Google Scholar 

  14. L. E. El’sgol’ts and S. B. Norkin, Introduction to the Theory of Differential Equations with Deviating Argument (Nauka, Moscow, 1971).

    MATH  Google Scholar 

  15. B. G. Grebenshchikov and A. B. Lozhnikov, “Stabilization methods for systems with two linear delays,” Russ. Math. 55, 15–23 (2011). https://doi.org/10.3103/S1066369X11090039

    Article  MATH  Google Scholar 

  16. B. G. Grebenshchikov and A. B. Lozhnikov, “Stabilization of the neutral-type system with two linear delays,” in Proc. 4th Int. Conf. on Mathematics, Its Applications, and Mathematical Education (Ulan-Ude, 2011), vol. 1, pp. 32–35.

  17. P. Lankaster, Theory of Matrices (Academic, New York, 1969).

    Google Scholar 

  18. E. M. Markushin, Optimal Automatic Control Systems with Time Delay (Saratovsk. Univ., Saratov, 1971).

    MATH  Google Scholar 

  19. R. Bellman and K. L. Cooke, Differential-Difference Equations, Ed. by R. Bellman, Mathematics in Science and Engineering, Vol. 6 (Academic, New York, 1963).

  20. B. G. Grebenshchikov, “The first-approxiamtion stability of a nonstationary system with delay,” Russ. Math. 56, 29–36 (2012). https://doi.org/10.3103/S1066369X12020041

    Article  MATH  Google Scholar 

  21. V. L. Kharitonov, “An extension of the prediction scheme to the case of systems with both input and state delay,” Automatica 50, 211–217 (2014). https://doi.org/10.1016/j.automatica.2013.09.042

    Article  MathSciNet  MATH  Google Scholar 

  22. B. Zhou, Q. Liu, and F. Mazenc, “Stabilization of linear systems with both input and state delays by observer–predictors,” Automatica 83, 368–377 (2017). https://doi.org/10.1016/j.automatica.2017.06.027

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Halanay and D. Wexler, Teoria calitativă a sistemelor cu impulsuri (Editura Acad. Republicii Socialiste Romania, Bucharest, 1968).

    MATH  Google Scholar 

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Authors and Affiliations

  1. South Ural State University, 454080, Chelyabinck, Russia

    B. G. Grebenshchikov

  2. Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, 620108, Yekaterinburg, Russia

    A. B. Lozhnikov

  3. Ural Federal University, 620002, Yekaterinburg, Russia

    A. B. Lozhnikov

Authors
  1. B. G. Grebenshchikov
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  2. A. B. Lozhnikov
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Correspondence to B. G. Grebenshchikov or A. B. Lozhnikov.

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The authors declare that they have no conflicts of interest.

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Translated by I. Tselishcheva

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Grebenshchikov, B.G., Lozhnikov, A.B. On the Choice of Methods for Stabilizing Systems Containing a Linear Delay. Russ Math. 67, 30–42 (2023). https://doi.org/10.3103/S1066369X23010036

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