Equivalence of control systems with linear systems on Lie groups and homogeneous spaces

Philippe Jouan

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 16, Issue: 4, page 956-973
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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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 16.4 (2010): 956-973. <http://eudml.org/doc/250747>.

@article{Jouan2010,
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 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. },
author = {Jouan, Philippe},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Lie groups; homogeneous spaces; linear systems; complete vector field; finite dimensional Lie algebra; Lie group; homogeneous space; linear system},
language = {eng},
month = {10},
number = {4},
pages = {956-973},
publisher = {EDP Sciences},
title = {Equivalence of control systems with linear systems on Lie groups and homogeneous spaces},
url = {http://eudml.org/doc/250747},
volume = {16},
year = {2010},
}

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
DA - 2010/10//
PB - EDP Sciences
VL - 16
IS - 4
SP - 956
EP - 973
AB - 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.
LA - eng
KW - Lie groups; homogeneous spaces; linear systems; complete vector field; finite dimensional Lie algebra; Lie group; homogeneous space; linear system
UR - http://eudml.org/doc/250747
ER -

References

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