Boundedness control sets for linear systems on Lie groups

Let Σ be a linear system on a connected Lie group G and assume that the reachable set A from the identity element e ∈ G is open. In this paper, we give an algebraic condition to warrant the boundedness of the existent control set with a nonempty interior containing e. We concentrate the search for the class of decomposable groups which includes any solvable group and also every compact semisimple group.