Skip to main content
SpringerLink
Log in

Methods for studying the asymptotic properties of systems with two linear delays

  • Ordinary Differential Equations
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

We study methods for the instability analysis of neutral-type systems with two delays linearly depending on the argument. In the case of instability of the neutral part of a system with constant coefficients, we suggest a stabilization algorithm. In addition, by using the Laplace transform, we obtain sufficient conditions for the instability of the solution of a given system.

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. Grebenshchikov, B.G., Stability of Stationary Systems of Neutral Type with Linear Delays, in Differentsial’nye uravneniya s chastnymi proizvodnymi (Partial Differential Equations), St. Petersburg, 1992, pp. 57–67.

    Google Scholar 

  2. Ockendon, J. and Taylor, A., The Dynamics of a Current Collection System for an Electric Locomotive, Proc. Roy. Soc. London Ser. A, 1971, vol. 322, pp. 447–468.

    Article  Google Scholar 

  3. Kolmanovskii, V.B. and Nosov, V.R., Ustoichivost’ i periodicheskie rezhimy reguliruemykh sistem s posledeistviem (Stability and Periodic Modes of Controlled Systems with Aftereffect), Moscow: Nauka, 1981.

    Google Scholar 

  4. Grebenshchikov, B.G., On Stabilization of Linear Systems with a Small Parameter Containing Linear Delay, J. Comput. System Sci., 2001, vol. 40, no. 5, pp. 679–688.

    MathSciNet  MATH  Google Scholar 

  5. Furasov, V.D., Ustoichivost’ i stabilizatsiya diskretnykh protsessov (Stability and Stabilization of Discrete Processes), Moscow: Nauka, 1982.

    Google Scholar 

  6. Krasovskii, N.N., Nekotorye zadachi teorii ustoichivosti dvizheniya (Certain Problems in the Theory of Stability of Motion), Moscow: Gosudarstv. Izdat. Fiz.-Mat. Lit., 1959.

    Google Scholar 

  7. El’sgol’ts, L.E. and Norkin, S.B., Vvedenie v teoriyu differentsial’nykh uravnenii s otklonyayushchimsya argumentom (Introduction to the Theory of Differential Equations with Deviating Argument), Moscow: Nauka, 1971.

    Google Scholar 

  8. Grebenshchikov, B.G., Stability of Systems with Variable Delay Linearly Depending on Time, in Ustoichivost’ i nelineinye kolebaniya (Stability and Nonlinear Oscillations), Sverdlovsk: Ural. Gos. Univ., 1983, pp. 25–33.

    Google Scholar 

  9. Bellman, R. and Cooke, K., Differential-Difference Equations, New York: Academic, 1963. Translated under the title Differentsial’no-raznostnye uravneniya, Moscow: Mir, 1967.

    Google Scholar 

  10. Markushevich, A.I., Kratkii kurs teorii analiticheskikh funktsii (A Short Course in the Theory of Analytic Functions), Moscow: Nauka, 1966.

    Google Scholar 

  11. Bellman, R., Introduction to Matrix Analysis, New York–Toronto–London: McGraw-Hill Book Co., 1960. Translated under the title Vvedenie v teoriyu matrits, Moscow: Nauka, 1969.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ural Federal University, Yekaterinburg, Russia

    B. G. Grebenshchikov & A. B. Lozhnikov

  2. Institute for Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, Russia

    B. G. Grebenshchikov & A. B. Lozhnikov

Authors
  1. B. G. Grebenshchikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. B. Lozhnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. B. Lozhnikov.

Additional information

Original Russian Text © B.G. Grebenshchikov, A.B. Lozhnikov, 2016, published in Differentsial’nye Uravneniya, 2016, Vol. 52, No. 1, pp. 3–19.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grebenshchikov, B.G., Lozhnikov, A.B. Methods for studying the asymptotic properties of systems with two linear delays. Diff Equat 52, 1–17 (2016). https://doi.org/10.1134/S0012266116010018

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266116010018

Keywords

Search

Navigation