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Probability-theoretical generalization of the second lyapunov method

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References

  1. H. Poincaré, Collected Works [Russian translation], Vol. 2, Nauka, Moscow (1972), pp. 130–158.

    Google Scholar 

  2. A. M. Lyapunov, The General Problem of Stability of Motion [in Russian], Gostekhizdat, Moscow-Leningrad (1950).

    Google Scholar 

  3. A. A. Voronov and V. M. Matrosov (eds.), The Method of Vector Lyapunov Functions in Stability Theory [in Russian], Nauka, Moscow (1987).

    Google Scholar 

  4. W. I. Zangwill, Nonlinear Programming: A Unified Approach, Prentice-Hall, Englewood Cliffs, NJ (1969).

    Google Scholar 

  5. Yu. G. Evtushenko and V. G. Zhadan, “Application of the Lyapunov function method to study the convergence of numerical methods,” Zh. Vychisl. Mat. Mat. Fiz.,15, No. 1, 101–112 (1975).

    Google Scholar 

  6. B. T. Polyak, An Introduction to Optimization [in Russian], Nauka, Moscow (1983).

    Google Scholar 

  7. Yu. I. Lyubich and G. D. Maistrovskii, “General theory of relaxation processes,” Usp. Mat. Nauk,25, No. 1, 57–112 (1970).

    Google Scholar 

  8. J. L. Doob, Stochastic Processes, Wiley, New York (1953).

    Google Scholar 

  9. R. S. Bucy, “Stability and positive supermartingales,” J. Diff. Eq.,1, No. 2, 151–155 (1965).

    Google Scholar 

  10. I. I. Gikhman and A. V. Skorokhod, Theory of Stochastic Processes [in Russian], Vol. 1, Nauka, Moscow (1971).

    Google Scholar 

  11. Yu. M. Ermol'ev, Stochastic Programming Methods [in Russian], Nauka, Moscow (1976).

    Google Scholar 

  12. A. N. Nakonechnyi, “Probability-theoretical generalization of the second Lyapunov method,” Dokl. Akad. Nauk UkrSSR, Ser. A, No. 2, 18–19 (1990).

    Google Scholar 

  13. J. Bernoulli, On the Law of Large Numbers [Russian translation], Nauka, Moscow (1986).

    Google Scholar 

  14. S. M. Ermakov, Monte-Carlo Method and Related Topics [in Russian], Nauka, Moscow (1975).

    Google Scholar 

  15. G. A. Mikhailov, Optimization of Weighted Monte-Carlo Methods [in Russian], Nauka, Moscow (1987).

    Google Scholar 

  16. I. N. Kovalenko and N. Yu. Kuznetsov, Computational Methods for High-Reliability Systems [in Russian], Radio i Svyaz’, Moscow (1988).

    Google Scholar 

  17. I. N. Kovalenko and A. N. Nakonechnyi, Approximate Computation and Optimization of Reliability [in Russian], Naukova Dumka, Kiev (1989).

    Google Scholar 

  18. V. D. Shpak, “Unbiased estimators for the solution of a linear integral equation of the second kind and their application to computation of reliability criteria of semi-Markov systems,” Dokl. Akad. Nauk UkrSSR, Ser. A, No. 10, 81–84 (1989).

    Google Scholar 

  19. M. Bazaraa and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, Wiley, New York (1979).

    Google Scholar 

  20. D. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York (1982).

    Google Scholar 

  21. J. Dennis and R. Schnabel, Numerical Methods for Unconstrained Minimization and Nonlinear Equations [Russian translation], Mir, Moscow (1988).

    Google Scholar 

  22. L. Armijo, “Minimization of functions having Lipschitz-continuous first partial derivatives,” Pacific J. Math.,16, 1–13 (1966).

    Google Scholar 

  23. G. I. Marchuk and Yu. A. Kuznetsov, “Iterative methods and quadratic functionals,” in: Methods of Computational Mathematics [in Russian], Nauka, Novosibirsk (1975), pp. 4–143.

    Google Scholar 

  24. Ya. Z. Tsypkin, A. S. Poznyak, and S. N. Tikhonov, “Optimal adaptive identification methods,” Itogi Nauki Tekh. Ser. Tekh. Kibern.,29, 3–44, VINITI, Moscow (1990).

    Google Scholar 

  25. Ya. Z. Tsypkin, A. V. Nazin, and A. S. Poznyak, “Adaptive finite systems,” Itogi Nauki i Tekh., Ser. Tekh. Kibern.,19, 69–141, VINITI, Moscow (1986).

    Google Scholar 

  26. A. V. Nazin, B. T. Polyak, and A. B. Tsybakov, “Passive stochastic approximation,” Avtom. Telemekh., No. 11, 127–134 (1989).

    Google Scholar 

  27. A. N. Nakonechnyi, “Extremal problems with rate events. II,” Kibernetika, No. 2, 41–48 (1992).

    Google Scholar 

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Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 68–81, January–February, 1993.

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Nakonechnyi, A.N. Probability-theoretical generalization of the second lyapunov method. Cybern Syst Anal 29, 53–62 (1993). https://doi.org/10.1007/BF01130089

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