no. 3
Invariant measures and controllability of finite systems on compact manifolds
ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 3, pp. 643-655.

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.

DOI : 10.1051/cocv/2011165
Classification : 17B66, 37A05, 37N35, 93B05, 93B17, 93C10
Mots clés : compact homogeneous spaces, linear systems, controllability, finite dimensional Lie algebras, Haar measure 
@article{COCV_2012__18_3_643_0,
     author = {Jouan, Philippe},
     title = {Invariant measures and controllability of finite systems on compact manifolds},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {643--655},
     publisher = {EDP-Sciences},
     volume = {18},
     number = {3},
     year = {2012},
     doi = {10.1051/cocv/2011165},
     mrnumber = {3041659},
     zbl = {1281.93020},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2011165/}
}
TY  - JOUR
AU  - Jouan, Philippe
TI  - Invariant measures and controllability of finite systems on compact manifolds
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2012
SP  - 643
EP  - 655
VL  - 18
IS  - 3
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2011165/
DO  - 10.1051/cocv/2011165
LA  - en
ID  - COCV_2012__18_3_643_0
ER  - 
%0 Journal Article
%A Jouan, Philippe
%T Invariant measures and controllability of finite systems on compact manifolds
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2012
%P 643-655
%V 18
%N 3
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2011165/
%R 10.1051/cocv/2011165
%G en
%F COCV_2012__18_3_643_0
Jouan, Philippe. Invariant measures and controllability of finite systems on compact manifolds. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 3, pp. 643-655. doi : 10.1051/cocv/2011165. http://www.numdam.org/articles/10.1051/cocv/2011165/

[1] H. Abbaspour and M. Moskowitz, Basic Lie Theory. World Scientific (1997). | Zbl

[2] M. Berger and B. Gostiaux, Géométrie différentielle  : variétés, courbes et surfaces. Presses universitaires de France (1987). | MR | Zbl

[3] A. Borel, Compact Clifford-Klein forms of symmetric spaces. Topology 2 (1963) 111-122. | MR | Zbl

[4] F. Cardetti and D. Mittenhuber, Local controllability for linear control systems on Lie groups. J. Dyn. Control Syst. 11 (2005) 353-373. | MR | Zbl

[5] J.P. Gauthier, Structure des systèmes non linéaires. Éditions du CNRS, Paris (1984). | MR | Zbl

[6] S. Helgason, Differential Geometry and Symmetric Spaces. Academic Press (1962). | MR | Zbl

[7] Ph. Jouan, Equivalence of control systems with linear systems on Lie groups and homogeneous spaces. ESAIM : COCV 16 (2010) 956-973. | Numdam | MR | Zbl

[8] Ph. Jouan, Finite time and exact time controllability on compact manifolds. J. Math. Sci. (to appear). | MR | Zbl

[9] V. Jurdjevic, Geometric control theory. Cambridge University Press (1997). | MR | Zbl

[10] V. Jurdjevic and H.J. Sussmann, Control systems on Lie groups. J. Differ. Equ. 12 (1972) 313-329. | MR | Zbl

[11] C. Lobry, Controllability of nonlinear systems on compact manifolds. SIAM J. Control. 12 (1974) 1-4. | MR | Zbl

[12] G.D. Mostow, Homogeneous spaces with finite invariant measure. Ann. Math. 75 (1962) 17-37. | MR | Zbl

[13] V.V. Nemytskii and V.V. Stepanov, Qualitative Theory of Differential Equations. Princeton Uniersity Press (1960). | MR | Zbl

[14] Yu.L. Sachkov, Controllability of invariant systems on Lie groups and homogeneous spaces. J. Math. Sci. 100 (2000) 2355-2427. | MR | Zbl

[15] L. San Martin and V. Ayala, Controllability properties of a class of control systems on Lie groups, Nonlinear control in the year 2000 1, Paris, Lecture Notes in Control and Inform. Sci. 258. Springer (2001) 83-92. | MR | Zbl

Cité par Sources :