Mathematical systems theory

, Volume 18, Issue 1, pp 125–133 | Cite as

Sur la Structure de l'Ensemble d'Accessibilité de Certains Systèmes: Application à l'Equivalence des Systèmes

  • G. Sallet
Article
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Abstract

For a large class of systems, the interior of the closure of the accessibility set from a point is the interior of this set. As a consequence, equivalent systems, in the Jurdjevic—Kupka's sense, have the same interior and the same boundary for their accessibility sets. This result is used to prove a conjecture of L. R. Hunt.

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Bibliographie

  1. 1.
    R. W. Brockett, Non linear systems and differential geometry,Proc. IEEE, 64, No. 1, 61–72 (1976).Google Scholar
  2. 2.
    J. Dieudonne,Foundations of Modern Analysis, Academic Press, New York, 1969.Google Scholar
  3. 3.
    R. Hermann and A. J. Krener, Non linear controllability and observability,IEEE Trans. Automat. Control., AC 22, No. 5, 728–740 (1977).Google Scholar
  4. 4.
    L. R. Hunt, Controllability of general linear systems,Math. System Theory, 12, 361–370 (1979).Google Scholar
  5. 5.
    L. R. Hunt, Global controllability of non linear systems in two dimensions,Math. Systems Theory, 13, 361–376 (1980).Google Scholar
  6. 6.
    L. R. Hunt, Controllability of non linear hypersurface systems, in: Algebtraic and geometric methods in linear systems theory,AMS Lectures in Applied Mathematics, 18, C. I. Byrnes and C. F. Martin, Eds., pp. 209–224.Google Scholar
  7. 7.
    L. R. Hunt,n-dimensional controllability with (n−1) controls,IEEE Trans. Automat. Control, AC 27, No. 1, 113–117 (1982).Google Scholar
  8. 8.
    V. Jurdjevic, Attainable sets and controllability; a geometric approach, in:Lectures Notes in Econom. and Math. Systems, No. 106, pp. 219–251.Google Scholar
  9. 9.
    V. Jurdjevic and I. Kupka, Control systems on semi-simple Lie groups and their homogeneous spaces,Ann. Inst. Fourier, 31, 4, 151–179 (1981).Google Scholar
  10. 10.
    V. Jurdjevic and I. Kupka, Control systems subordinated to a group action: Accessibility,J. Differential Equations, 39, No. 2, 186–211 (1981).Google Scholar
  11. 11.
    H. Kunita, On the controllability of non-linear systems,Appl. Math Optim., 5, 89–99 (1979).Google Scholar
  12. 12.
    I. Kupka and G. Sallet, A sufficient condition for the transitivity of pseudo-semi-groups. Application to system theory,J. Differential Equations, to appear.Google Scholar
  13. 13.
    C. Lobry, Controlabilité des systèmes non lineaires,SIAM J. Control, 8, 573–605 (1970).Google Scholar
  14. 14.
    C. Lobry, Bases mathématiques de la théorie du contrôle, cours de troisième cycle. Multigraphié, Bordeaux (1978).Google Scholar
  15. 15.
    C. Lobry and P. Brunovsky, Controlabilité Bang-Bang, controlabilité differentiable, et perturbations des systèmes linéaires,Ann. Mat. Appl., sér. 4, 55, 93–119 (1975).Google Scholar
  16. 16.
    H. J. Sussmann, Orbits of families of vector fields and integrability of distributions,Trans. Amer. Math. Soc., 180, 171–188 (1973).Google Scholar
  17. 17.
    H. J. Sussmann and V. Jurdjevic, Controllability of nonlinear systems,J. Differential Equations, 00, 95–116 (1972).Google Scholar
  18. 18.
    A. Bacciotti and G. Stephani, The region of attainability of nonlinear system with unbounded controls,J. Optimization Theory and Applications, 35, 1, 57–84 (1981).Google Scholar
  19. 19.
    A. Bacciotti and G. Stephani, On the relationship between global and local controllability,Math. Systems Theory, 16, 79–91 (1983).Google Scholar

Copyright information

© Springer-Verlag New York Inc 1985

Authors and Affiliations

  • G. Sallet
    • 1
  1. 1.Laboratoire de méthodes mathématiques d'analyse des systèmesERA-CNRS 040399-Université de MetzMetzFrance

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