Existence and Uniqueness of Control Sets with a Nonempty Interior for Linear Control Systems on Solvable Groups

In this paper, we establish weak conditions for the existence of a control set with nonempty interior for a linear control system on a solvable Lie group. We show that the Lie algebra rank condition, together with the compactness of the nilpotent part of the generalized kernel of the drift, is sufficient to guarantee the existence of such a control set. Moreover, this control set is unique and contains the entire generalized kernel in its closure.