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.