Soviet Applied Mechanics

, Volume 15, Issue 10, pp 901–918 | Cite as

The Lyapunov-function method in stability theory of motion of complex system

  • A. A. Martynyuk
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Literature Cited

  1. 1.
    L. Yu. Anapol'skii, V. D. Irtegov, and V. M. Matrosov, “Methods of constructing Lyapunov functions,” in: Results in Science and Technology. General Mechanics [in Russian], Vol. 2 VINITI, Moscow (1975), pp. 53–112.Google Scholar
  2. 2.
    E. A. Barbashin, Lyapunov Functions [in Russian], Nauka, Moscow (1970).Google Scholar
  3. 3.
    N. N. Bogolyubov, “Vibrations,” in: Mechanics in the USSR in the Past Thirty Years [in Russian], Moscow (1950), pp. 99–114.Google Scholar
  4. 4.
    N. N. Bogolyubov (Bogoliubov) and Yu. A. Mitropol'skii (I. A. Mitropolski), Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach (1962).Google Scholar
  5. 5.
    G. S. Vakhonina, A. S. Zemlyakov, and V. M. Matrosov, “Methods of constructing quadratic Lyapunov vector-functions for linear systems,” Avtomat. Telemekh., No. 2, 5–16 (1973).Google Scholar
  6. 6.
    E. A. Grebennikov and Yu. A. Ryabov, New Qualitative Methods in Celestial Mechanics [in Russian], Nauka, Moscow (1971).Google Scholar
  7. 7.
    N. P. Erugin, “The first Lyapunov method,” in: Mechanics in the USSR in the Past Fifty Years [in Russian], Moscow (1968), pp. 67–867.Google Scholar
  8. 8.
    V. F. Zadorozhnyi and A. A. Martynyuk, “Estimates of the effect of coupling between subsystems on stability of a linear nonstationary system,” Prikl. Mekh.,8, No. 9, 65–71 (1972).Google Scholar
  9. 9.
    V. F. Zadorozhnyi and A. A. Martyunuk, “Solution of the general aggregation problem as a moment problem,” Mat. Fiz., No. 14, 49–55 (1973).Google Scholar
  10. 10.
    V. F. Zadorozhnyi and A. A. Martyunyuk, “The direct Lyapunov method and the L-moment problem in stability problems of multidimensional systems,” in: Problems of Analytic Mechanics, Stability Theory, and Control [in Russian], Nauka, Moscow (1975), pp. 151–154.Google Scholar
  11. 11.
    V. F. Zadorozhnyi, E. A. Nurminskii, and A. Yu. Stanishevskii, “Minimax approach to synthesis problems of multidimensional dynamical systems,” Kibernetika, No. 5, 111–114 (1977).Google Scholar
  12. 12.
    A. S. Zemlyakov, “Construction of a comparison system,” Tr. Kazan. Aviats. Inst. No. 144, 46–54 (1972).Google Scholar
  13. 13.
    V. I. Zubov, “Stability of multicomponent periodic motions,” Differents. Uravn.,8, No. 9, 1693–1694 (1972).Google Scholar
  14. 14.
    N. N. Krasovskii, Stability of Motion, Stanford Univ. Press (1963).Google Scholar
  15. 15.
    P. A. Kuz'min, “History of domestic schools of stability of motion,” in: Problems in Analytical Mechanics, Stability Theory and Control [in Russian], Nauka, Moscow (1975), pp. 3–11.Google Scholar
  16. 16.
    A. I. Kukhtenko, “Basic problems in control theory by complex systems,” Proc. Seminar Complex Control Systems [in Russian], Inst. Kibern. Akad. Nauk UkrSSR, No. 1, pp. 3–40 (1968).Google Scholar
  17. 17.
    V. A. Lazaryan, L. A. Dlugach, and M. L. Korotenko, Stability of Motion of Rail Crews [in Russian], Nauka Dumka, Kiev (1972).Google Scholar
  18. 18.
    A. M. Lyapunov, “General problem of stability of motion,” in: Collected Works of Lyapunov [in Russian], Vol. 2, Moscow (1956), pp. 7–264.Google Scholar
  19. 19.
    A. M. Lyapunov (Liapunov), “An investigation of one of the singular cases of the theory of stability of motion,” in: Stability of Motion, Academic Press (1966).Google Scholar
  20. 20.
    I. G. Malkin, “Stability of constantly acting perturbations,” Prikl. Mat. Mekh.,8, No. 3, 241–245 (1944).Google Scholar
  21. 21.
    A. A. Martyunyuk, “Technical stability of complex systems,” in: Cybernetics and Computational Technology: Complex Control Systems [in Russian], No. 15 (1972), pp. 58–64.Google Scholar
  22. 22.
    A. A. Martynyuk, “The instability of equilibrium positions of a multidimensional system consisting of ‘neutrally’ unstable subsystems,” Prikl. Mekh.,8, No. 6, 77–82 (1972).Google Scholar
  23. 23.
    A. A. Martynyuk, “The stability of multidimensional systems,” in: Analytical and Qualitative Methods of the Theory of Differential Equations [in Russian], Inst. Mat. Akad. Nauk UkrSSR, Kiev (1972), pp. 158–174.Google Scholar
  24. 24.
    A. A. Martynyuk, Technical Stability in Dynamics [in Russian], Tekhnika, Kiev (1973).Google Scholar
  25. 25.
    A. A. Martynyuk, “Stability of a multidimensional system at a finite interval,” in: Cybernetics and Computational Technology: Complex Control Systems [in Russian], No. 19, Kiev (1973), pp. 79–83.Google Scholar
  26. 26.
    A. A. Martynyuk and V. I. Kosolapov, “The comparison principle and the averaging method in the stability problem of nonasymptotically stable motion for constantly acting perturbations,” Preprint No. 78-33, Inst. Mat. Akad. Nauk UkrSSR (1978).Google Scholar
  27. 27.
    A. A. Martynyuk, “Decomposition and aggregation in system analysis,” Teor. Mech.,1, 87–93 (1975).Google Scholar
  28. 28.
    A. A. Martynyuk, Stability of Motion of Complex Systems [in Russian], Naukova Dumka, Kiev (1975).Google Scholar
  29. 29.
    A. A. Martynyuk, “Qualitative and numerical-analytic studies of stability of motion,” Prikl. Mekh.,13, No. 10, 87–93 (1977).Google Scholar
  30. 30.
    A. A. Martynyuk, “A theorem of the comparison principle in nonlinear mechanics,” Prikl. Mekh.,14, No. 10, 129–132 (1978).Google Scholar
  31. 31.
    A. A. Martynyuk and R. Gutovski, Integral Inequalities and Stability of Motion [in Russian], Naukova Dumka, Kiev (1979).Google Scholar
  32. 32.
    A. A. Martynyuk and A. Yu. Obolenskii, “Stability studies of autonomic comparison systems,” Preprint No. 78-28, Inst. Mat. Akad. Nauk UkrSSR, Kiev (1978).Google Scholar
  33. 33.
    V. M. Matrosov, “The method of Lyapunov vector functions in systems with inverse coupling,” Avtomat. Telemekh., No. 9, 63–75 (1972).Google Scholar
  34. 34.
    V. M. Matrosov, “The method of Lyapunov vector functions in analysis of complex systems with parameter distributions,” Avtomat. Telemekh., No. 1, 5–21 (1973).Google Scholar
  35. 5.
    G. I. Mel'nikov, Dynamics of Nonlinear Mechanical and Electromechanical Systems [in Russian], Mashinostroenie, Leningrad (1975).Google Scholar
  36. 36.
    Yu. A. Mitropol'skii and A. A. Martynyuk, “Certain directions of investigations into stability of periodic motions and the theory of nonlinear vibrations,” Prikl. Mekh.,14, No. 3, 3–13 (1978).Google Scholar
  37. 37.
    A. A. Piontkovskii and L. D. Rutkovskaya, “Study of some problems in the theory of stability by the method of Lyapunov vector functions,” Avtomat. Telemekh., No. 10, 23–32 (1967).Google Scholar
  38. 38.
    V. L. Rvachev, Geometric Applications of Logic Algebra [in Russian], Tekhnika, Kiev (1967).Google Scholar
  39. 39.
    B. B. Rumyantsev, “The Lyapunov function method in the theory of stability of motion,” in: Mechanics in the USSR in the Past Fifty Years [in Russian], Moscow (1968), pp. 7–66.Google Scholar
  40. 40.
    A. Ya. Savchenko, Stability of Stationary Motion of Mechanical Systems [in Russian], Naukova Dumka, Kiev (1977).Google Scholar
  41. 41.
    M. M. Khapaev, “A theorem of the Lyapunov type,” Dokl. Akad. Nauk SSSR176, No. 6, 1262–1265 (1967).Google Scholar
  42. 42.
    S. A. Chaplygin, “A new method of approximate integration of differential equation,” in: Selected Works [in Russian], Moscow (1976), pp. 307–360.Google Scholar
  43. 43.
    N. G. Chetaev (Chetayev), The Stability of Motion, Pergamon Press (1961).Google Scholar
  44. 44.
    F. N. Bailey, “The application of Lyapunov's second method to interconnected systems,” J. Soc. Ind. Appl. Math., Ser. A, Control, No. 3, 443–462 (1965).Google Scholar
  45. 45.
    L. T. Grujic and D. D. Siljak, “Stability of large-scale systems with stable and unstable subsystems,” IACC Conf., Stanford (August, 1972), pp. 550–555.Google Scholar
  46. 46.
    L. T. Grujic, “Stability analysis of large-scale systems with stable and unstable systems,” Int. J. Control20, No. 3, 453–463 (1974).Google Scholar
  47. 47.
    L. T. Grujic, “Vector Lyapunov functions and singularly perturbed large-scale systems,” Proc. Joint Autom. Control, New York (1976), pp. 408–415.Google Scholar
  48. 48.
    L. T. Grujic and D. D. Siljak, “Asymptotic stability and instability of large-scale systems,” IEE Trans. Autom. Control,AC-18, No. 6, 636–645 (1973).Google Scholar
  49. 49.
    A. A. Martynyuk, “Technical stability of nonlinear and control systems,” Nonlinear Vibr. Probl., No. 19, 21–84 (1979).Google Scholar
  50. 50.
    V. M. Matrosov, “Vector Lyapunov functions in the analysis of nonlinear interconnected systems,” Symposia Mathematica, VI, Bologna (1971), pp. 209–242.Google Scholar
  51. 51.
    A. N. Michel, “Stability analysis of interconnected systems,” Berlin Math. Stat. Sec. Forschungszentr. Graz., No. 4, 1–107 (1973).Google Scholar
  52. 52.
    A. N. Michel, “Quantitative analysis of simple and interconnected systems: stability, boundedness, and trajectory behavior,” IEEE Trans. Circuit Theory,CT-17, No. 3, 292–301 (1970).Google Scholar
  53. 53.
    A. N. Michel, “Stability analysis of interconnected systems,” SIAM J. Control,12, No. 3, 554–579 (1974).Google Scholar
  54. 54.
    M. Rao, “Upper and lower bounds of the norm of solutions of nonlinear Volterra integral equations,” Proc. Nat. Acad. Sci., No. 33, 263–266 (1963).Google Scholar
  55. 55.
    L. Weiss and E. F. Infante, “Finite time stability under perturbing forces and product spaces,” IEEE Trans. Autom. Control.AC-12, No. 1, 54–59 (1967).Google Scholar

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© Plenum Publishing Corporation 1980

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  • A. A. Martynyuk

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