Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Development of the direct Lyapunov method for delay systems (survey)

  • 46 Accesses

  • 2 Citations

This is a preview of subscription content, log in to check access.

Literature cited

  1. 1.

    N. N. Krasovskii, Stability of Motion. Applications of Lyapunov's Second Method to Differential Equations with Delay, Stanford University Press, Stanford, California (1963).

  2. 2.

    A. M. Lyapunov (M. A. Liapounoff), Probléme Général de Stabilité du Mouvement, Princeton Univ. Press, Princeton (1947).

  3. 3.

    A. A. Martynyuk, Technical Stability in Dynamics [in Russian], Tekhnika, Kiev (1973).

  4. 4.

    A. A. Martynyuk, Stability of Motion of Composite Systems [in Russian], Naukova Dumka, Kiev (1975).

  5. 5.

    A. D. Myshkis, Linear Differential Equations with Retarded Argument [in Russian], Nauka, Moscow (1972).

  6. 6.

    B. S. Razumikhin, "On the stability of systems with delay," Prikl. Mat. Mekh.,20, No. 5, 500–512 (1956).

  7. 7.

    J. K. Hale, Theory of Functional Differential Equations, 2nd edn., Springer, New York (1977).

  8. 8.

    T. A. Burton, "Uniform asymptotic stability in functional differential equations," Proc. Am. Math. Soc.,68, No. 3, 195–199 (1978).

  9. 9.

    T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, Orlando (1985).

  10. 10.

    T. Burton and L. Hatvani, "Stability theorems for nonautonomous functional-differential equations by Lyapunov functionals," Tôhoku Math. J.,41, No. 1, 65–104 (1989).

  11. 11.

    L. E. El'sgolts, Introduction to the Theory of Differential Equations with Deviating Arguments, Holden-Day, San Francisco (1966).

  12. 12.

    A. Halanay, Differential Equations. Stability, Oscillations, Time Lags, Academic Press, New York (1966).

  13. 13.

    J. Kato, "On Lyapunov-Razumikhin type theorems for functional differential equations," Funkcial. Ekvac.,16, No. 2, 225–239 (1973).

  14. 14.

    J. Kato, "Lyapunov's second method in functional differential equations," Tôhoku Math. J.,32, No. 4, 487–497 (1980).

  15. 15.

    V. Lakshmikantham, "Recent advances in the stability theory of nonlinear systems," in: Qualitative Theory of Differential Equations (Szeged, 1988), Colloq. Math, Soc. János Bolyai, 53, North-Holland, Amsterdam (1990), pp. 383–393.

  16. 16.

    V. Lakshmikantham and S. Leela, Differential and Integral Inequalities. Theory and Applications, Vol. 1: Ordinary Differential Equations, Academic Press, New York (1969).

  17. 17.

    V. Lakshmikantham and S. Leela, "On perturbing Lyapunov functions," Math. Systems Theory,10, No. 1, 85–90 (1976).

  18. 18.

    V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Dekker, New York (1989).

  19. 19.

    V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Practical Stability of Nonlinear Systems, World Scientific, Singapore (1990).

  20. 20.

    V. Lakshmikantham, and S. Leela, and S. Sivasundaram, "Lyapunov functions on product spaces and stability theory of delay differential equations," J. Math. Anal. Appl.,154, No. 3, 391–402 (1991).

  21. 21.

    V. Lakshmikantham and Xin Zhi Liu, "Perturbing families of Lyapunov functions and stability in terms of two meausres," J. Math. Anal. Appl.,140, No. 1, 107–114 (1989).

  22. 22.

    Xin Zhi Liu, "Stability in terms of two meausres for functional differential equations," J. Differential Integral Equations, No. 3, 257–261 (1989).

  23. 23.

    G. R. Shendge, "A new approach to the stability theory of functional differential systems," J. Math. Anal. Appl.,95, No. 3, 319–334 (1983).

  24. 24.

    T. Yoshizawa, Stability Theory by Lyapunov's Second Method, the Mathematical Society of Japan, Tokyo (1966).

Download references

Additional information

Florida Institute of Technology, Melbourne, Florida, USA. Institute of Mechanics of the Academy of Sciences of the Ukraine, Kiev (Ukraine). Translated from Prikladnaya Mekhanika, Vol. 29, No. 2, pp. 3–16, February, 1993.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Lakshmikantham, V., Martynyuk, A.A. Development of the direct Lyapunov method for delay systems (survey). Int Appl Mech 29, 83–96 (1993). https://doi.org/10.1007/BF00846981

Download citation

Keywords

  • Delay System
  • Lyapunov Method
  • Direct Lyapunov Method