A semigroup associated to a linear control system on a Lie group

Let us consider a linear control system Σ on a connected Lie group G. It is known that the accessibility set A from the identity e is in general not a semigroup. In this article we associate a new algebraic object SΣ to Σ which turns out to be a semigroup, allowing the use of the semigroup machinery to approach Σ. In particular, we obtain some controllability results.