Attainable sets for left invariant control systems and   Carnot-Caratheodory metrics on nilpotent Lie groups
  Let $G$ be a semidirect product of a simply connected nilpotent Lie group and $\R$. For a left invariant control system on $G$ with a convex cone as a control domain, it is proved that the attainable sets coincides with a "halfspace" if the degree of contact of the with some special subspaces is sufficiently high. 
V.M.Gichev