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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  1. V. Ayala and L. San Martin, Controllability properties of a class of control systems on Lie groups, in Nonlinear control in the year 2000, Vol. 1 (Paris), Lect. Notes Control Inform. Sci.258, Springer (2001) 83–92.  
  2. V. Ayala and J. Tirao, Linear control systems on Lie groups and Controlability, in Proceedings of Symposia in Pure Mathematics, Vol. 64, AMS (1999) 47–64.  
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  5. G. Hochschild, The Structure of Lie Groups. Holden-Day (1965).  
  6. Ph. Jouan, On the existence of observable linear systems on Lie Groups. J. Dyn. Control Syst.15 (2009) 263–276.  
  7. V. Jurdjevic, Geometric control theory. Cambridge University Press (1997).  
  8. P. Malliavin, Géométrie différentielle intrinsèque. Hermann, Paris, France (1972).  
  9. 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.  
  10. R.S. Palais, A global formulation of the Lie theory of transformation groups, Memoirs of the American Mathematical Society22. AMS, Providence, USA (1957).  
  11. 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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