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Equivalence of control systems with linear systems on 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
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Abstract

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

A vector field on a connected Lie group is linear if its flow is a one parameter group of automorphisms. An affine vector field is obtained by adding a left 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 with the projections of right invariant vector fields.

A linear system on a homogeneous space is a system whose drift part is affine 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

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