Journal of Dynamical and Control Systems

, Volume 11, Issue 3, pp 353–373 | Cite as

Local Controllability for Linear Control Systems on Lie Groups

Original Article

Abtract

In this work, we study controllability properties of linear control systems on Lie groups. Using Lie theory of semigroups, we obtain local controllability results for this type of systems. In addition, properties for the flow of the drift vector field X and for the reachable set of the system are presented. Finally, an example on the Heisenberg Lie group is considered, and its properties are proved using the theory developed.

Key words and phrases.

Systems on Lie groups noninvariant systems linear control systems local controllability 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    1. V. Ayala and L. San Martin, Controllability properties of a class of control systems on Lie groups. In: Nonlinear control in the year 2000, Vol. 1 (Paris), Lect. Notes Control Inform. Sci. 258 Springer, London (2001), pp. 83–92.Google Scholar
  2. 2.
    2. V. Ayala and J. Tirao, Linear control systems on Lie groups and local controllability. In: Differential geometry and control (G. Ferreyra, R. Gardner, H. Hermes, and H. Sussmann, Eds.), Amer. Math. Soc., Providence, Rhode Island (1999), pp. 47–64.Google Scholar
  3. 3.
    3. N. Bourbaki, Lie groups and Lie algebras. Chapters 1–3. Springer-Verlag, Berlin (1998).Google Scholar
  4. 4.
    4. J. Dixmier, L'application exponentielle dans les groupes de Lie re;aasolubles. Bull. Soc. Math. France 85 (1957), pp. 113–121.MATHMathSciNetGoogle Scholar
  5. 5.
    5. J. Hilgert, K. H. Hofmann, and J. D. Lawson, Lie groups, convex cones, and semigroups. The Clarendon Press Oxford University Press, New York (1989).Google Scholar
  6. 6.
    6. J. Hilgert and K.-H. Neeb, Lie semigroups and their applications. Springer-Verlag, Berlin (1993).Google Scholar
  7. 7.
    7. J. D. Lawson and D. Mittenhuber, Controllability of Lie systems. In: Contemporary trends in nonlinear geometric control theory and its applications (A. Anzaldo-Meneses, F. Monroy Perez, B. Bonnard, and J. P. Gauthier, eds.), World Scientific Press (2002), pp. 53–76.Google Scholar
  8. 8.
    8. L. Markus, Controllability of multitrajectories on Lie groups. In: Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980), Springer-Verlag, Berlin (1981), pp. 250–265.Google Scholar
  9. 9.
    9. F. Warner, Foundations of differentiable manifolds and Lie groups. Springer-Verlag, New York (1983).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ConnecticutStorrsUSA
  2. 2.Rauser Towers Perrin GmbHReutlingenGermany

Personalised recommendations