International Applied Mechanics

, Volume 47, Issue 1, pp 86–96 | Cite as

Revisiting the theory of stability on time scales of the class of linear systemswith structural perturbations

  • S. V. Babenko

Linear systems of dynamic equations with periodic coefficients and structural perturbations on time scale are analyzed for Lyapunov stability. Sufficient conditions for the asymptotic stability of the equations are established based on the matrix-value concept of Lyapunov’s direct method for all values of the structural matrix from the structural set. A system of two dynamic equations on time scale is considered as an example of applying the theoretical results obtained


asymptotic stability structural perturbations time scale dynamic equations Lyapunov’s direct method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. V. Babenko and V. I. Slyn’ko, “Stability of the solutions of dynamic equations based on discontinuous functions,” Dop. NAN Ukrainy, No. 9, 7–12 (2009).Google Scholar
  2. 2.
    Lj. T. Grujic, A. A. Martynyuk, and M. Ribbens-Pavella, Large-Scale Systems Stability under Structural and Singular Perturbations, Springer-Verlag, Berlin (1987).MATHCrossRefGoogle Scholar
  3. 3.
    P. Lancaster, Theory of Matrices, Acad. Press, New York–London (1978).Google Scholar
  4. 4.
    A. A. Martynyuk and V. I. Slyn’ko, “Setting up a matrix-valued Lyapunov function for a linear periodic system on time scale,” Dokl. RAN, 417, No.1, 18–22 (2007).MathSciNetGoogle Scholar
  5. 5.
    Yu. A. Martynyuk-Chernienko, “Stability of dynamic systems on time scale,” Dokl. RAN, 413, No. 1, 1–5 (2007).Google Scholar
  6. 6.
    S. V. Babenko, “Stability conditions for a class of dynamic equations describing discrete continuous motion of mechanical systems,” Int. Appl. Mech., 45, No. 3, 315 (2009).Google Scholar
  7. 7.
    M. Bohner and A. A. Martynyuk, “Elements of Lyapunov stability theory for dynamic equations on time scale,” Int. Appl. Mech., 43, No. 9, 949–970 (2007).ADSCrossRefGoogle Scholar
  8. 8.
    M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications, Birkhaser, Boston (2001).MATHGoogle Scholar
  9. 9.
    S. Hilger, “Analysis on measure chains: a unified approach to continuous and discrete calculus,” Res. in Math., 18, 18–56 (1990).MathSciNetMATHGoogle Scholar
  10. 10.
    T. A. Luk’yanova, “Stability and boundedness conditions for the motions of discrete-time mechanical systems,” Int. Appl. Mech., 45, No. 8, 917–921 (2009).CrossRefGoogle Scholar
  11. 11.
    A. A. Martynyuk and A. S. Khoroshun, “On parametric asymptotic stability of large-scale systems,” Int. Appl. Mech., 44, No. 5, 565–574 (2008).ADSCrossRefGoogle Scholar
  12. 12.
    A. A. Martynyuk and V. I. Slyn’ko, “Stability results for large-scale difference systems via matrix-valued Liapunov functions,” Nonlin. Dynam. Syst. Theor., 7, No. 2, 217–224 (2007).MathSciNetMATHGoogle Scholar
  13. 13.
    D. D. Siljak, Large-Scale Dynamic Systems: Stability and Structure, North-Holland, New York (1978).MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  1. 1.S. P. Timoshenko Institute of MechanicsNational Academy of Sciences of UkraineKyivUkraine

Personalised recommendations