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Cybernetics and Systems Analysis

, Volume 44, Issue 5, pp 673–679 | Cite as

Controlled dynamic systems and Carleman operator

  • V. F. Zadorozhny
Article

Abstract

Controlled dynamic systems on a compact set and thus with invariant measure are considered. This allows reducing differential dynamic systems to Fredholm-type integral equations. An algorithm of constructing a vector field of maximum descent rate along a trajectory is presented. The algorithm is reduced to a numerical moment-type procedure.

Keywords

dynamic system compact set control stability Hilbert-Schmidt and Carleman operators measure 

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References

  1. 1.
    V. I. Arnol’d, Ordinary Differential Equations [in Russian], Nauka, Moscow (1971).Google Scholar
  2. 2.
    I. Tamura, Topology of Foliations, Transl. Math. Monographs, 97, American Math. Soc., Rhode Island (1992).MATHGoogle Scholar
  3. 3.
    A. Povzner, “Global existence theorem for a nonlinear system and deficiency index of a linear operator,” Sib. Mat. Zh., 5, No. 2, 377–386 (1964).MathSciNetGoogle Scholar
  4. 4.
    P. R. Halmos and V. S. Sander, Bounded Integral Operators on L 2 Spaces, Springer-Verlag (1979).Google Scholar
  5. 5.
    N. N. Krasovskii, Problems of Stabilization of Controlled Motion, Supplement IV to the monograph I. G. Malkin, Theory of Motion Stability [in Russian], Nauka, Moscow (1966).Google Scholar
  6. 6.
    V. I. Zubov, Mathematical Methods for Analysis of Automatic Control Systems [in Russian], Mashinostroenie, Leningrad (1974).Google Scholar
  7. 7.
    N. N. Bogolyubov, Complete Works [in Russian], Vol. 2, Naukova Dumka, Kyiv (1976).Google Scholar
  8. 8.
    K. Moren, Methods in Hilbert Space [Russian translation], Mir, Moscow (1965).Google Scholar
  9. 9.
    I. M. Gel’fand and A. G. Kostychenko, “Eigenfunction expansion of differential and other operators,” Dokl. AN SSSR, 103, No. 3, 349–352 (1955).MATHGoogle Scholar
  10. 10.
    V. I. Zubov, “Integral equations for the Lyapunov function,” Dokl. AN SSSR, 314, No. 4, 780–782 (1990).Google Scholar
  11. 11.
    I. G. Petrovskii, Lectures on the Theory of Integral Equations [in Russian], Nauka, Moscow (1965).Google Scholar
  12. 12.
    M. G. Krein and A. A. Nudel’man, Markov Moment Problem and Extremum Problems [in Russian], Nauka, Moscow (1978).Google Scholar
  13. 13.
    V. F. Zadorozhnyi, “The Lyapunov problem in dynamic control systems,” Cybern. Syst. Analysis, 38, No. 6, 904–910 (2002).CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.V. M. Glushkov Institute of CyberneticsNational Academy of Sciences of UkraineKyivUkraine

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