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

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

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

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topJouan, 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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- G. Hochschild, The Structure of Lie Groups. Holden-Day (1965).
- Ph. Jouan, On the existence of observable linear systems on Lie Groups. J. Dyn. Control Syst.15 (2009) 263–276.
- V. Jurdjevic, Geometric control theory. Cambridge University Press (1997).
- P. Malliavin, Géométrie différentielle intrinsèque. Hermann, Paris, France (1972).
- L. Markus, Controllability of multitrajectories on Lie groups, in Dynamical systems and turbulence, Warwick (1980), Lect. Notes Math.898, Springer, Berlin-New York (1981) 250–265.
- R.S. Palais, A global formulation of the Lie theory of transformation groups, Memoirs of the American Mathematical Society22. AMS, Providence, USA (1957).
- H.J. Sussmann, Orbits of families of vector fields and integrability of distributions. Trans. Amer. Math. Soc.180 (1973) 171–188.

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