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Absolute Stability of General Systems

  • P. M. Salzberg
Part of the NATO Conference Series book series (NATOCS, volume 5)

Abstract

In a fundamental paper, J. Auslander and P. Seibert made an exhaustive study of prolongations in the sense of T. Ura (Cf. [12] and [13]) and their connection with the second method of Liapunov [2]. The context was that of a dynamical system on a locally compact metric space. Ever since, and with varying success, attempts to improve these results have been made in different directions. A noteworthy contribution is due to O. Hájek [4], who extended the characterization of absolute stability given in [2], to the case of closed sets in a paracompact locally compact space endowed with a dynamical system.

Keywords

Topological Space Ordinal Number Absolute Stability Transitive Relation Liapunov Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    J. Auslander, “Generalized Recurrence in Dynamical Systems, Contributions to Differential Equations,” 3, 65–74, 1964.Google Scholar
  2. 2.
    J. Auslander, and P. Seibert, “Prolongations and Stability in Dynamical Systems,” Ann. Inst. Fourier, Grenoble, 14, pp. 237–267, 1964.CrossRefGoogle Scholar
  3. 3.
    J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, 1968.Google Scholar
  4. 4.
    O. Hájek, “Absolute Stability of Noncompact Sets,” Journ. Diff. Equations, 9, pp. 496–508, 1971.CrossRefGoogle Scholar
  5. 5.
    L. Nachbin, Topology and Order, D. Van Nostrand Company, Inc., 1965.Google Scholar
  6. 6.
    E. Roxin, “Stability in General Control System,” Journ. of Diff. Equations, 1, pp. 115–150, 1965.CrossRefGoogle Scholar
  7. 7.
    P. M. Salzberg, “On the Existence of Continuous and Semi-Continuous Liapunov Functions,” Funkcial. Ekvac., 19, pp. 19–26, 1976.Google Scholar
  8. 8.
    P. M. Salzberg, and P. Seibert, “A Necessary and Sufficient Condition for the Existence of a Liapunov Function,” Funkcial. Ekvac., 16, pp. 97–101, 1973.Google Scholar
  9. 9.
    P. M. Salzberg, and P. Seibert, “Remarks on a Universal Criterion for Liapunov Stability,” Funkcial. Ekvac., 18, pp. 1–4, 1975.Google Scholar
  10. 10.
    P. M. Salzberg, and P. Seibert, “A Unified Theory of Attraction in General Systems,” Techn. Report DS 76–1, Dpto. de Mat. y Ci. Comp. Universidad Simon Bolivar, Caracas, Venezuela, 1976.Google Scholar
  11. 11.
    P. Seibert, “A Unified Theory of Liapunov Stability,” Funkcial. Ekvac., 15, pp. 139–147, 1972.Google Scholar
  12. 12.
    T. Ura, “Sur les courbes définies par les equations différentielles dans L’espace à m dimensions,” Ann. Sci. Ecole Norm. Sup., (3), 70, pp. 287–360, 1953.Google Scholar
  13. 13.
    T. Ura, “Sur le courant extérieur à une région invariante,” Funkcial. Ekvac., 2, pp. 143–200, 1959.Google Scholar

Copyright information

© Springer Science+Business Media New York 1978

Authors and Affiliations

  • P. M. Salzberg
    • 1
  1. 1.Departamento de MatemáticasUniversidad Simon BolívarCaracasVenezuela

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