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

Philippe Jouan (1)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem

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 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 have its own interest. The present proof makes use of geometric control theory arguments.

Mots clés: Lie groups; Homogeneous spaces; Linear systems; Complete vector field; Finite dimensional Lie algebra

Domaines: Optimisation et contrôle [math.OC]; Géométrie différentielle [math.DG]

Philippe Jouan. Equivalence of Control Systems with Linear Systems on Lie Groups and Homogeneous Spaces. 2008. ⟨hal-00342749⟩