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

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-00342749
Contributor : Philippe Jouan Connect in order to contact the contributor
Submitted on : Friday, November 28, 2008 - 1:43:05 PM
Last modification on : Tuesday, October 19, 2021 - 4:13:30 PM
Long-term archiving on: : Monday, June 7, 2010 - 11:32:05 PM

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  • HAL Id : hal-00342749, version 1
  • ARXIV : 0812.0058

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

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