Russian Mathematics

, Volume 62, Issue 4, pp 60–73 | Cite as

Absolute Logarithmic Norm

  • A. I. Perov
  • I. D. Kostrub
  • O. I. Kleshchina
  • E. E. Dikarev
Article
  • 2 Downloads

Abstract

In this paper we introduce and study a new concept of the absolute logarithmic norm, which has much in common with the classical definition of the logarithmic norm by S. M. Lozinskii. The the theory that we develop allows to obtain new facts from the Lyapunov stability theory for the systems of linear differential equations with constant coefficients. The presentation of the material relies heavily on the theory of off-diagonally nonnegative matrices arising from the Perron–Frobenius theory for nonnegative matrices.

Keywords

logarithmic norm Lyapunov stability the theory of Perron–Frobenius for nonnegative matrices 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Faddeev, D. K., Faddeeva, V. N. Computational Methods of Linear Algebra (Fizmatgiz, Moscow–Leningrad, 1963) [in Russian].MATHGoogle Scholar
  2. 2.
    Daletskii, Yu. L., Krein, M. G. Stability of Solutions of Differential Equations in Banach Space (Nauka, Moscow, 1970) [in Russian].Google Scholar
  3. 3.
    Bylov, B. F., Vinograd, R. E., Grobman, D. M., Nemytskii, V. V. Theory of Lyapunov Exponents and its Application to Problems of Stability (Nauka, Moscow, 1966) [in Russian].Google Scholar
  4. 4.
    Lozinskii, S. M. “Error Estimate for Numerical Integration of Ordinary Differential Equations. I”, Izv. Vyssh. Uchebn. Zaved.Mat., No. 5, 52–90 (1958) [in Russian]MATHGoogle Scholar
  5. 5.
    Beckenbach E.F., Bellman R. Inequalities (Springer, Berlin–Gö ttingen–Heidelberg, 1961; Mir, Moscow, 1965).CrossRefMATHGoogle Scholar
  6. 6.
    Kamke E. Handbook of Ordinary Differential Equations (Nauka, Moscow, 1971) [Russian translation].Google Scholar
  7. 7.
    Matrosov, V. M. Lyapunov Vector Functions Method: Analysis of Dynamical Properties of Nonlinear Systems (Fizmatlit, Moscow, 2001). [Russian].Google Scholar
  8. 8.
    Perov, A. I. “New Features of Stability of Linear Systems of Differential Equations With Constant Coefficients”, RussianMathematics 58, No. 9, 41–48 (2014).MATHGoogle Scholar
  9. 9.
    Demidovich, B. P. Lectures on the Mathematical Stability Theory (Nauka, Moscow, 1967) [in Russian].MATHGoogle Scholar
  10. 10.
    Sansone G. Ordinary Differential Equations (In. Lit.,Moscow, 1954), Vol. 2 [Russian translation].Google Scholar
  11. 11.
    Sevast’yanov, B. A. “Theory of Branching Stochastic Processes”, Usp. Mat. Nauk 6, No. 6, 47–99 (1951) [in Russian].Google Scholar
  12. 12.
    Gantmacher, F. R. Theory of Matrices (Nauka, Moscow, 1967) [in Russian].MATHGoogle Scholar
  13. 13.
    Perov, A. I., Kostrub, I. D., Avdeeva, O. I. “Asymptotic Stability Criteria”, in Proceedings of International conference dedicated to the 95-th anniversary of Voronezh university ‘Topical Problems of Applied Mathematic, Information Science and Mechanics’, (Nauchnaya Kniga Voronezh, 2013), pp. 21–30 [in Russian].Google Scholar
  14. 14.
    Kamenskii, M., Nistri, P. “An Averaging Method for Singulary Perturbed System of Semilinear Differetial Inclusions With C0-Semigroups”, Set-Valued Analysis 11, No. 4, 345–357 (2003).MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Bellman R. Introduction to Matrix Analysis (McGraw-Hill Book Co., Inc., 1960, New York–Toronto–London; Nauka, Moscow, 1969).MATHGoogle Scholar
  16. 16.
    Krasnosel’skii, M. A., Burd, V. Sh., Kolesov. Yu. S. Nonlinear Almost Periodic Oscillations (Nauka, Moscow, 1970) [in Russian].Google Scholar
  17. 17.
    Glazman, I. M., Lyubich, Yu. I. Finite-Dimensional Linear Analysis (Nauka,Moscow, 1969) [in Russian].MATHGoogle Scholar
  18. 18.
    Yakubovich, V. A., Starzhinskii, V. M. Linear Differential EquationsWith Periodic Coefficients and Their Applications (Nauka, Moscow, 1972) [in Russian].Google Scholar
  19. 19.
    Markus, M., Mink, Kh. A Survey of Matrix Theory and Matrix Inequalities (Nauka, Moscow, 1972) [in Russian].Google Scholar
  20. 20.
    Parodi, M. Localization of Characteristic Numbers of Matrices and its Applications (Academic, N. Y., 1959; In. Lit.,Moscow, 1960).Google Scholar

Copyright information

© Allerton Press, Inc. 2018

Authors and Affiliations

  • A. I. Perov
    • 1
  • I. D. Kostrub
    • 1
  • O. I. Kleshchina
    • 1
  • E. E. Dikarev
    • 1
  1. 1.Voronezh State UniversityVoronezhRussia

Personalised recommendations