Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-13T03:38:56.207Z Has data issue: false hasContentIssue false
  • English
  • Français

Equivalence of control systems with linear systemson Lie groups and homogeneous spaces

Published online by Cambridge University Press:  31 July 2009

Philippe Jouan
Philippe Jouan*
Affiliation:
LMRS, CNRS UMR 6085, Université de Rouen, avenue de l'Université, BP 12, 76801 Saint-Étienne-du-Rouvray, France. Philippe.Jouan@univ-rouen.fr
Get access

Abstract

The aim of this paper is to prove that a control affine system on a manifold is equivalent by diffeomorphismto a linear system on a Lie group or a homogeneous space if and only if the vectorfields of the system are complete and generate a finite dimensionalLie algebra.

A vector field on a connected Lie group is linear if its flow is a one parametergroup of automorphisms. An affine vector field is obtained by adding aleft invariant one. Its projection on a homogeneous space, whenever it exists, is still called affine.

Affine vector fields on homogeneous spaces can be characterized by their Lie brackets withthe projections of right invariant vector fields.

A linear system on a homogeneous space is a system whose drift part isaffine and whose controlled part is invariant.

The main result is based on a general theorem on finite dimensional algebras generated by complete vector fields, closely related to a theorem of Palais, and which has its own interest. The present proof makes use of geometric control theory arguments.

Keywords

Lie groupshomogeneous spaceslinear systemscomplete vector fieldfinite dimensional Lie algebra
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 4 , October 2010 , pp. 956 - 973
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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 (2001) 83–92.
V. Ayala and J. Tirao, Linear control systems on Lie groups and Controlability, in Proceedings of Symposia in Pure Mathematics, Vol. 64, AMS (1999) 47–64.
N. Bourbaki, Groupes et algèbres de Lie, Chapitres 2 et 3. CCLS, France (1972).
Cardetti, F. and Mittenhuber, D., Local controllability for linear control systems on Lie groups. J. Dyn. Control Syst. 11 (2005) 353–373. CrossRef
G. Hochschild, The Structure of Lie Groups. Holden-Day (1965).
Ph. Jouan, On the existence of observable linear systems on Lie Groups. J. Dyn. Control Syst. 15 (2009) 263–276. CrossRef
V. Jurdjevic, Geometric control theory. Cambridge University Press (1997).
P. Malliavin, Géométrie différentielle intrinsèque. Hermann, Paris, France (1972).
L. Markus, Controllability of multitrajectories on Lie groups, in Dynamical systems and turbulence, Warwick (1980), Lect. Notes Math. 898, Springer, Berlin-New York (1981) 250–265.
R.S. Palais, A global formulation of the Lie theory of transformation groups, Memoirs of the American Mathematical Society 22. AMS, Providence, USA (1957).
Sussmann, H.J., Orbits of families of vector fields and integrability of distributions. Trans. Amer. Math. Soc. 180 (1973) 171–188. CrossRef