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, p. 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 : https://doi.org/10.1051/cocv/2009027
Classification:  17B66,  57S15,  57S20,  93B17,  93B29
Keywords: Lie groups, homogeneous spaces, linear systems, complete vector field, finite dimensional Lie algebra
@article{COCV_2010__16_4_956_0,
     author = {Jouan, Philippe},
     title = {Equivalence of control systems with linear systems on Lie groups and homogeneous spaces},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {16},
     number = {4},
     year = {2010},
     pages = {956-973},
     doi = {10.1051/cocv/2009027},
     zbl = {pre05821904},
     mrnumber = {2744157},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2010__16_4_956_0}
}
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://www.numdam.org/item/COCV_2010__16_4_956_0/

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