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The aim of this paper is to prove that a control affine system on a manifoldis equivalent by diffeomorphism to a linear system on a Lie group or ahomogeneous space if and only the vector fields of the system are complete andgenerate a finite dimensional Lie algebra. A vector field on a connected Liegroup is linear if its flow is a one parameter group of automorphisms. Anaffine vector field is obtained by adding a left invariant one. Its projectionon a homogeneous space, whenever it exists, is still called affine. Affinevector fields on homogeneous spaces can be characterized by their Lie bracketswith the projections of right invariant vector fields. A linear system on ahomogeneous space is a system whose drift part is affine and whose controlledpart is invariant. The main result is based on a general theorem on finitedimensional algebras generated by complete vector fields, closely related to atheorem of Palais, and which have its own interest. The present proof makes useof geometric control theory arguments.

Equivalence of Control Systems with Linear Systems on Lie Groups and Homogeneous Spaces