Semigroup actions on homogeneous spaces: control sets and stabilizer subgroups

Given a control system on a manifold we investigate the properties of the reachable set from an arbitrarily chosen point. We are especially interested in the following questions: which points lie in the interior of their reachable set and which points are stabilizable in the sense that one can find a trajectory-defined on a nontrivial interval-starting and terminating at that point? We will investigate these questions in a Lie group setting, using the links between control theory and semigroup theory.