Invariant measures and controllability of finite systems on compact manifolds

Philippe Jouan

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 3, page 643-655
  • ISSN: 1292-8119

Abstract

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A control system is said to be finite if the Lie algebra generated by its vector fields is finite dimensional. Sufficient conditions for such a system on a compact manifold to be controllable are stated in terms of its Lie algebra. The proofs make use of the equivalence theorem of [Ph. Jouan, ESAIM: COCV 16 (2010) 956–973]. and of the existence of an invariant measure on certain compact homogeneous spaces.

How to cite

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Jouan, Philippe. "Invariant measures and controllability of finite systems on compact manifolds." ESAIM: Control, Optimisation and Calculus of Variations 18.3 (2012): 643-655. <http://eudml.org/doc/277822>.

@article{Jouan2012,
abstract = {A control system is said to be finite if the Lie algebra generated by its vector fields is finite dimensional. Sufficient conditions for such a system on a compact manifold to be controllable are stated in terms of its Lie algebra. The proofs make use of the equivalence theorem of [Ph. Jouan, ESAIM: COCV 16 (2010) 956–973]. and of the existence of an invariant measure on certain compact homogeneous spaces. },
author = {Jouan, Philippe},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Compact homogeneous spaces; linear systems; controllability; finite dimensional Lie algebras; Haar measure; compact homogeneous spaces},
language = {eng},
month = {11},
number = {3},
pages = {643-655},
publisher = {EDP Sciences},
title = {Invariant measures and controllability of finite systems on compact manifolds},
url = {http://eudml.org/doc/277822},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Jouan, Philippe
TI - Invariant measures and controllability of finite systems on compact manifolds
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/11//
PB - EDP Sciences
VL - 18
IS - 3
SP - 643
EP - 655
AB - A control system is said to be finite if the Lie algebra generated by its vector fields is finite dimensional. Sufficient conditions for such a system on a compact manifold to be controllable are stated in terms of its Lie algebra. The proofs make use of the equivalence theorem of [Ph. Jouan, ESAIM: COCV 16 (2010) 956–973]. and of the existence of an invariant measure on certain compact homogeneous spaces.
LA - eng
KW - Compact homogeneous spaces; linear systems; controllability; finite dimensional Lie algebras; Haar measure; compact homogeneous spaces
UR - http://eudml.org/doc/277822
ER -

References

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  1. H. Abbaspour and M. Moskowitz, Basic Lie Theory. World Scientific (1997).  Zbl1137.17001
  2. M. Berger and B. Gostiaux, Géométrie différentielle  : variétés, courbes et surfaces. Presses universitaires de France (1987).  Zbl0619.53001
  3. A. Borel, Compact Clifford-Klein forms of symmetric spaces. Topology2 (1963) 111–122.  Zbl0116.38603
  4. F. Cardetti and D. Mittenhuber, Local controllability for linear control systems on Lie groups. J. Dyn. Control Syst.11 (2005) 353–373.  Zbl1085.93004
  5. J.P. Gauthier, Structure des systèmes non linéaires. Éditions du CNRS, Paris (1984).  
  6. S. Helgason, Differential Geometry and Symmetric Spaces. Academic Press (1962).  Zbl0111.18101
  7. Ph. Jouan, Equivalence of control systems with linear systems on Lie groups and homogeneous spaces. ESAIM : COCV16 (2010) 956–973.  Zbl1210.93024
  8. Ph. Jouan, Finite time and exact time controllability on compact manifolds. J. Math. Sci. (to appear).  Zbl0877.93011
  9. V. Jurdjevic, Geometric control theory. Cambridge University Press (1997).  Zbl0940.93005
  10. V. Jurdjevic and H.J. Sussmann, Control systems on Lie groups. J. Differ. Equ.12 (1972) 313–329.  Zbl0237.93027
  11. C. Lobry, Controllability of nonlinear systems on compact manifolds. SIAM J. Control.12 (1974) 1–4.  Zbl0286.93006
  12. G.D. Mostow, Homogeneous spaces with finite invariant measure. Ann. Math.75 (1962) 17–37.  Zbl0115.25702
  13. V.V. Nemytskii and V.V. Stepanov, Qualitative Theory of Differential Equations. Princeton Uniersity Press (1960).  Zbl0089.29502
  14. Yu.L. Sachkov, Controllability of invariant systems on Lie groups and homogeneous spaces. J. Math. Sci.100 (2000) 2355–2427.  Zbl1073.93511
  15. L. San Martin and V. Ayala, Controllability properties of a class of control systems on Lie groups, Nonlinear control in the year 20001, Paris, Lecture Notes in Control and Inform. Sci.258. Springer (2001) 83–92.  

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