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.

Classification: 17B66, 57S15, 57S20, 93B17, 93B29

Keywords: Lie groups, homogeneous spaces, linear systems, complete vector field, finite dimensional Lie algebra

@article{COCV_2010__16_4_956_0, author = {Jouan, Philippe}, title = {Equivalence of control systems with linear systems on Lie groups and homogeneous spaces}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, publisher = {EDP-Sciences}, volume = {16}, number = {4}, year = {2010}, pages = {956-973}, doi = {10.1051/cocv/2009027}, zbl = {pre05821904}, mrnumber = {2744157}, language = {en}, url = {http://www.numdam.org/item/COCV_2010__16_4_956_0} }

Jouan, Philippe. Equivalence of control systems with linear systems on Lie groups and homogeneous spaces. ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, pp. 956-973. doi : 10.1051/cocv/2009027. http://www.numdam.org/item/COCV_2010__16_4_956_0/

[1] 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.

and ,[2] Linear control systems on Lie groups and Controlability, in Proceedings of Symposia in Pure Mathematics, Vol. 64, AMS (1999) 47-64. | Zbl 0916.93015

and ,[3] Groupes et algèbres de Lie, Chapitres 2 et 3. CCLS, France (1972). | Zbl 0483.22001

,[4] Local controllability for linear control systems on Lie groups. J. Dyn. Control Syst. 11 (2005) 353-373. | Zbl 1085.93004

and ,[5] The Structure of Lie Groups. Holden-Day (1965). | Zbl 0131.02702

,[6] On the existence of observable linear systems on Lie Groups. J. Dyn. Control Syst. 15 (2009) 263-276. | Zbl 1203.93050

,[7] Geometric control theory. Cambridge University Press (1997). | Zbl 1138.93005

,[8] Géométrie différentielle intrinsèque. Hermann, Paris, France (1972). | Zbl 0282.53001

,[9] Controllability of multitrajectories on Lie groups, in Dynamical systems and turbulence, Warwick (1980), Lect. Notes Math. 898, Springer, Berlin-New York (1981) 250-265. | Zbl 0515.34054

,[10] A global formulation of the Lie theory of transformation groups, Memoirs of the American Mathematical Society 22. AMS, Providence, USA (1957). | Zbl 0178.26502

,[11] Orbits of families of vector fields and integrability of distributions. Trans. Amer. Math. Soc. 180 (1973) 171-188. | Zbl 0274.58002

,