Skip to main content

Advertisement

Lyapunov functions for linear nonautonomous dynamical equations on time scales

  • 888 Accesses

  • 8 Citations

Abstract

The existence of a Lyapunov function is established following a method of Yoshizawa for the uniform exponential asymptotic stability of the zero solution of a nonautonomous linear dynamical equation on a time scale with uniformly bounded graininess.

[1234567891011121314151612345678910111213141516]

References

  1. 1.

    Agarwal RP: Difference Equations and Inequalities, Monographs and Textbooks in Pure and Applied Mathematics. Volume 155. Marcel Dekker, New York; 1992:xiv+777.

  2. 2.

    Agarwal RP, Bohner M, O'Regan D, Peterson A: Dynamic equations on time scales: a survey. Journal of Computational and Applied Mathematics 2002,141(1–2):1–26. 10.1016/S0377-0427(01)00432-0

  3. 3.

    Aulbach B, Hilger S: Linear dynamic processes with inhomogeneous time scale. In Nonlinear Dynamics and Quantum Dynamical Systems (Gaussig, 1990), Math. Res.. Volume 59. Akademie, Berlin; 1990:9–20.

  4. 4.

    Bohner M, Peterson A: Dynamic Equations on Time Scales. Birkhäuser Boston, Massachusetts; 2001:x+358.

  5. 5.

    Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser Boston, Massachusetts; 2003:xii+348.

  6. 6.

    Conway JB: Functions of One Complex Variable I, Graduate Texts in Mathematics. Volume 11. 2nd edition. Springer, New York; 1978:xiii+317.

  7. 7.

    Döffinger A: Theorie dynamischer Gleichungen—ein einheitlicher Zugang zur kontinuierlichen und diskreten Dynamik, Diplomarbeit. Universität Augsburg, Augsburg; 1995.

  8. 8.

    Hilger S: Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Dissertation. Universität Würzburg, Würzburg; 1988.

  9. 9.

    Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics 1990,18(1–2):18–56.

  10. 10.

    Hilger S: Special functions, Laplace and Fourier transform on measure chains. Dynamic Systems and Applications 1999,8(3–4):471–488.

  11. 11.

    Keller S: Asymptotisches Verhalten invarianter Faserbündel bei Diskretisierung und Mittelwertbildung im Rahmen der Analysis auf Zeitskalen, Dissertation. Universität Augsburg, Augsburg; 1999.

  12. 12.

    Kloeden PE, Khilger S: The effect of time granularity on the asymptotic stability of dynamical systems. Automation and Remote Control 1994,55(9, part 1):1293–1298 (1995).

  13. 13.

    Pötzsche C, Siegmund S, Wirth F: A spectral characterisation of exponential stability for linear time-invariant systems on time scales. Discrete and Continuous Dynamical Systems 2002, 9: 255–265.

  14. 14.

    Remmert R: Funktionentheorie. Springer, Berlin; 1995.

  15. 15.

    Yoshizawa T: Stability Theory by Liapunov's Second Method, Publications of the Mathematical Society of Japan, no. 9. The Mathematical Society of Japan, Tokyo; 1966:viii+223.

  16. 16.

    Zmorzynska A: Lyapunovfunktionen auf Zeitskalen, Diplomarbeit. J. W. Goethe Universität, Frankfurt am Main; 2004.

Download references

Author information

Correspondence to Peter E Kloeden.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Kloeden, P.E., Zmorzynska, A. Lyapunov functions for linear nonautonomous dynamical equations on time scales. Adv Differ Equ 2006, 069106 (2006) doi:10.1155/ADE/2006/69106

Download citation

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Functional Equation