Journal of Dynamical and Control Systems

, Volume 20, Issue 1, pp 91–104 | Cite as

Semigroups of Simple Lie Groups and Controllability



In this paper, we consider a subsemigroup S of a real connected simple Lie group G generated by {exp tX : X ∈ Γ, t ≥ 0} for some subset Γ of L, the Lie algebra of G. It is proved that for an open class Γ = {A, ± B} and a generic pair (A, B) in L × L, if S contains a subgroup isomorphic to SL(2, ℝ), associated to an arbitrary root, then S is the whole G. In a series of previous papers, analogous results have been obtained for the maximal root only. Recently, a similar result for complex connected simple Lie groups was proved. The proof uses special root properties that characterize some particular subalgebras of L. In control theory, this case Γ = {A, ± B} is specially important since the control system, ġ = (A + uB)g, where u ∈ ℝ, is controllable on G if and only if S = G.


Simple Lie groups Invariant vector fields Root systems Controllability 

AMS Subject Classification (2010)

93B05 93C10 17B22 22E46 


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Laboratoire de MathématiquesINSA de RouenSaint-Etienne-du-RouvrayFrance

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