Equivalence of control systems with linear systems on Lie groups and homogeneous spaces
ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, pp. 956-973.

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.

DOI: 10.1051/cocv/2009027
Classification: 17B66, 57S15, 57S20, 93B17, 93B29
Keywords: Lie groups, homogeneous spaces, linear systems, complete vector field, finite dimensional Lie algebra
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Jouan, Philippe. Equivalence of control systems with linear systems on Lie groups and homogeneous spaces. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, pp. 956-973. doi : 10.1051/cocv/2009027. http://archive.numdam.org/articles/10.1051/cocv/2009027/

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