Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-22T21:08:33.339Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant measures and controllability of finite systems oncompact manifolds

Published online by Cambridge University Press:  14 September 2011

Philippe Jouan
Philippe Jouan*
Affiliation:
Lab. R. Salem, CNRS UMR 6085, Université de Rouen, avenue de l’Université, BP 12, 76801 Saint-Étienne-du-Rouvray, France. [email protected]
Get access

Abstract

A control system is said to be finite if the Lie algebra generated by its vector fieldsis finite dimensional. Sufficient conditions for such a system on a compact manifold to becontrollable are stated in terms of its Lie algebra. The proofs make use of theequivalence theorem of [Ph. Jouan, ESAIM: COCV 16 (2010)956–973]. and of the existence of an invariant measure on certain compact homogeneousspaces.

Keywords

Compact homogeneous spaceslinear systemscontrollabilityfinite dimensional Lie algebrasHaar measure
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 3 , July 2012 , pp. 643 - 655
Copyright
© EDP Sciences, SMAI, 2011

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

H. Abbaspour and M. Moskowitz, Basic Lie Theory. World Scientific (1997).
M. Berger and B. Gostiaux, Géométrie différentielle  : variétés, courbes et surfaces. Presses universitaires de France (1987).
Borel, A., Compact Clifford-Klein forms of symmetric spaces. Topology 2 (1963) 111–122. Google Scholar
Cardetti, F. and Mittenhuber, D., Local controllability for linear control systems on Lie groups. J. Dyn. Control Syst. 11 (2005) 353–373. Google Scholar
J.P. Gauthier, Structure des systèmes non linéaires. Éditions du CNRS, Paris (1984).
S. Helgason, Differential Geometry and Symmetric Spaces. Academic Press (1962).
Jouan, Ph., Equivalence of control systems with linear systems on Lie groups and homogeneous spaces. ESAIM : COCV 16 (2010) 956–973. Google Scholar
Ph. Jouan, Finite time and exact time controllability on compact manifolds. J. Math. Sci. (to appear).
V. Jurdjevic, Geometric control theory. Cambridge University Press (1997).
Jurdjevic, V. and Sussmann, H.J., Control systems on Lie groups. J. Differ. Equ. 12 (1972) 313–329. Google Scholar
Lobry, C., Controllability of nonlinear systems on compact manifolds. SIAM J. Control. 12 (1974) 1–4. Google Scholar
Mostow, G.D., Homogeneous spaces with finite invariant measure. Ann. Math. 75 (1962) 17–37. Google Scholar
V.V. Nemytskii and V.V. Stepanov, Qualitative Theory of Differential Equations. Princeton Uniersity Press (1960).
Sachkov, Yu.L., Controllability of invariant systems on Lie groups and homogeneous spaces. J. Math. Sci. 100 (2000) 2355–2427. Google Scholar
L. San Martin and V. Ayala, Controllability properties of a class of control systems on Lie groups, Nonlinear control in the year 2000 1, Paris, Lecture Notes in Control and Inform. Sci. 258. Springer (2001) 83–92.