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.
