Symmetries and Local Controllability

  • P. E. Crouch
  • C. I. Byrnes
Part of the Mathematics and Its Applications book series (MAIA, volume 29)


In this paper we review some results concerning the local controllability of nonlinear control systems. We stress those results which are most closely related to the existence of certain symmetries, including results by the authors and H. J. Sussmann. We also comment on the relation between this work and generalizations of Lie group theory to semigroups and Lie wedges.


Vector Field Local Controllability Homogeneous Element Closed Convex Cone Nonlinear Control System 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bianchini and G. Stef ani, “Normal Local Controllability of Order One” — preprint, Facolta’ di Ingegnevia Universita di Firenze.Google Scholar
  2. 2.
    P. Brunovsky, “Local Controllability of Odd Systems,” Banach Center Publications, Warsaw, Poland, Vol. 1, (1974), pp. 39–45.Google Scholar
  3. 3.
    C. Chevalley, The Theory of Lie Groups, Princeton University Press, Princeton Mathematical Series No. 8, (1946).Google Scholar
  4. 4.
    P. E. Crouch and C. I. Byrnes, “Local Accessibility, Local Reachability, and Representations of Compact Groups,” preprint, Department of Electrical Engineering, Arizona State University, Tempe (1985).Google Scholar
  5. 5.
    H. J. Sussmann, “A Sufficient Condition for Local Controllability,” S.I.A.M. J. Control, Vol. 16, (1978), pp. 790–802.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    P. E. Crouch and F. Lamnabhi, “Local Controllability About a Reference Trajectory,” Proc. 24th C.D.C./I.E.E.E. meeting Fort Lauderdale 1985.Google Scholar
  7. 7.
    P. E. Crouch, “Solvable Approximations to Control Systems,” S.I.A.M. J. Control, Vol. 22, (1984), pp. 40–54.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    R. W. Goodmann, Nil potent Lie Groups, Structure and Applications to Analysis, Lecture Notes in Mathematics, No. 562, Springer Verlag, 1976.Google Scholar
  9. 9.
    H. Hermes, “Control Systems Which Generate Decomposable Lie Algebras,” J. Diff. Egns. Vol. 44, pp. 166–187 (1982).MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    K. H. Hofmann and J. D. Lawson, “On Sophus Lie’s Fundamental Theorems, I,” Indag. Math., Vol. 45, (1983), pp. 453–466.MathSciNetzbMATHGoogle Scholar
  11. 11.
    K. H. Hofmann and J. D. Lawson, “On Sophus Lie’s Fundamental Theorems, II,” to appear in Indag. Math. (1982).Google Scholar
  12. 12.
    K. H. Hofmann and J. D. Lawson, “Foundations of Lie Semigroup,” Lecture Notes in Mathematics, Vol. 998, Springer Verlag, (1983), pp. 128–201.MathSciNetCrossRefGoogle Scholar
  13. 13.
    J. Hilgert and K. H. Hofmann, “Lie Theory for Semigroups,” Semigroup Forum, Vol. 30, Springer Verlag, (1984), pp. 243–251.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    A. J. Krener, “On the Equivalence of Control Systems and the Linearization of Nonlinear Systems,” S.I.A.M. J. on Control and Optimization, Vol. 15, (1977), pp. 813–829.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    A. J. Krener, “A Generalization of Chow’s Theorem and the Bang-Bang Theorem to Nonlinear Control Systems,” S.I.A.M. J. Control, Vol. 12, (1974), pp. 43–52.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    L. P. Rothschild and Stein, “Hypoelliptic Differential Operations and Nil potent Groups,” Acta. Math., Vol. 137, pp. 247–320 (1976).MathSciNetCrossRefGoogle Scholar
  17. 17.
    G. Stefani, “On Local Controllability and Related Topics,” preprint, Facolta di Ingegneria Universita di Firenze.Google Scholar
  18. 18.
    G. Stef ani, “Local Properties of Nonlinear Control Systems,” preprint, Facolta di Ingegneria Universita di Firenze.Google Scholar
  19. 19.
    H. J. Sussmann, “Lie Brackets and Local Controllability: A Sufficient Condition for Scalar-Input Systems,” S.I.A.M. J. of Control Vol. 21, (1983), pp. 686–713.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    H. J. Sussmann, “A General Theorem on Local Controllability,” preprint, Mathematics Dept. Rutgers University, New Brunswick, (1985).Google Scholar
  21. 21.
    H. J. Sussmann and V. Jurdjevic, “Controllability of Nonlinear Systems,” J. Diff. Eqns. Vol. 12, (1972), pp. 95–116.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    J. Hilgert and K. H. Hofmann, “On Sophus Lie’s Fundamental Theorems,” preprint, Technische Hochschule Darmstodt, (1984).Google Scholar
  23. 23.
    J. Hilgert, K. N. Hofmann and J. D. Lawson, “Controllability of Systems on a Nilpotent Lie Group,” Beitrage zur Algebra and Geometric, Vol. 20, (1985), pp. 185–190.MathSciNetzbMATHGoogle Scholar

Copyright information

© D. Reidel Publishing Company 1986

Authors and Affiliations

  • P. E. Crouch
    • 1
  • C. I. Byrnes
    • 1
  1. 1.Department of Electrical and Computer EngineeringArizona State UniversityTempeUSA

Personalised recommendations