Skip to main content
SpringerLink
Log in

Spectral Test for the Exponential Stability

  • ORDINARY DIFFERENTIAL EQUATIONS
  • Published:
Differential Equations Aims and scope Submit manuscript

Abstract

Using the theory of commutative Banach algebras, we establish an estimate for the solutions of a linear homogeneous matrix differential equation whose coefficients pairwise commute. A test for asymptotic stability in the sense of Lyapunov is derived from the resulting estimate.

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. Strelkov, S.P., Vvedenie v teoriyu kolebanii (Introduction to Vibration Theory), Moscow: Nauka, 1964.

    Google Scholar 

  2. Petrovskii, I.G., Lektsii po teorii obyknovennykh differentsial’nykh uravnenii (Lectures on the Theory of Ordinary Differential Equations), Moscow: Fizmatlit, 1984.

    Google Scholar 

  3. Kurosh, A.G., Kurs vysshei algebry (A Course in Higher Algebra), Moscow: Nauka, 1968.

    Google Scholar 

  4. Pontryagin, A.S., Obyknovennye differentsial’nye uravneniya (Ordinary Differential Equations), Moscow: Nauka, 1965.

    MATH  Google Scholar 

  5. Gantamkher, F.R., Teoriya matrits (Theory of Matrices), Moscow: Nauka, 1983.

    Google Scholar 

  6. Kharitonov, V.L., On the asymptotic stability of the equilibrium of a family of systems of linear differential equations, Differ. Uravn., 1978, vol. 14, no. 11, pp. 2086–2088.

    MathSciNet  Google Scholar 

  7. Kharitonov, V.L., The Routh–Hurwitz problem for a family of polynomials and quasipolynomials, Mat. Fiz., 1979, no. 26, pp. 69–79.

  8. Faedo, S., Un nuova problema di stabilita per le equazione algebriche a coefficienti reali, Ann. Scuola Norm. Super. Piza, 1953, vol. 7, no. 1–2, pp. 53–63.

    MathSciNet  MATH  Google Scholar 

  9. Kurbatov, V.G. and Kurbatova, I.V., Vychislitel’nye metody spektral’noi teorii (Computational Methods of Spectral Theory), Voronezh: Nauchn. Kniga, 2019.

    Google Scholar 

  10. Rudin, W., Functional Analysis, New York: McGraw-Hill, 1973. Translated under the title: Funktsional’nyi analiz, Moscow: Mir, 1975.

    MATH  Google Scholar 

  11. Lyusternik, L.A. and Sobolev, V.I., Kratkii kurs funktsional’nogo analiza (A Short Course in Functional Analysis), Moscow: Vyssh. Shkola, 1982.

    MATH  Google Scholar 

  12. Demidovich, B.P., Lektsii po matematicheskoi teorii ustoichivosti (Lectures on the Mathematical Theory of Stability), Moscow: Nauka, 1967.

    Google Scholar 

  13. Perov, A.I. and Kostrub, I.D., Differential equations in Banach algebras, Dokl. Math., 2020, vol. 101, pp. 139–143.

    Article  Google Scholar 

Download references

Funding

A.I. Perov’s studies were financially supported by the Russian Foundation for Basic Research, project no. 19–01–00732.

Author information

Authors and Affiliations

  1. Voronezh State University, Voronezh, 394018, Russia

    A. I. Perov & I. D. Kostrub

Authors
  1. A. I. Perov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. D. Kostrub
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. I. Perov or I. D. Kostrub.

Additional information

Translated by V. Potapchouck

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Perov, A.I., Kostrub, I.D. Spectral Test for the Exponential Stability. Diff Equat 57, 1565–1569 (2021). https://doi.org/10.1134/S0012266121120028

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0012266121120028

Search

Navigation