Let G be a semisimple Lie group and let S subset-of G be a subsemigroup with nonempty interior. In this paper we study invariant control sets for the action of S on homogeneous spaces of G. These sets on the boundary manifolds of the group are characterized in terms of the semisimple elements contained in int S. From this characterization a result on controllability of control systems on semisimple Lie groups is derived. Invariant control sets for the action of S on the boundaries of larger groups GBAR with G subset-of GBAR are also studied. This latter case includes the action of S on the projective space and on the flag manifolds.614161