Equivalence of control systems with linear systems on Lie groups and homogeneous spaces
LMRS, CNRS UMR 6085, Université
de Rouen, avenue de l'Université, BP 12, 76801
Saint-Étienne-du-Rouvray, France. Philippe.Jouan@univ-rouen.fr
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.
Mathematics Subject Classification: 17B66 / 57S15 / 57S20 / 93B17 / 93B29
Key words: Lie groups / homogeneous spaces / linear systems / complete vector field / finite dimensional Lie algebra
© EDP Sciences, SMAI, 2009