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.
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},
pages = {956--973},
year = {2010},
publisher = {EDP Sciences},
volume = {16},
number = {4},
doi = {10.1051/cocv/2009027},
mrnumber = {2744157},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2009027/}
}
TY - JOUR AU - Jouan, Philippe TI - Equivalence of control systems with linear systems on Lie groups and homogeneous spaces JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2010 SP - 956 EP - 973 VL - 16 IS - 4 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2009027/ DO - 10.1051/cocv/2009027 LA - en ID - COCV_2010__16_4_956_0 ER -
%0 Journal Article %A Jouan, Philippe %T Equivalence of control systems with linear systems on Lie groups and homogeneous spaces %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 956-973 %V 16 %N 4 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2009027/ %R 10.1051/cocv/2009027 %G en %F 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, Tome 16 (2010) no. 4, pp. 956-973. doi: 10.1051/cocv/2009027
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