An important tool in optimal control theory is the Pontrjagin maximum principle. A necessary condition for optimality, the principle is an analytic description of a control whose response stays in the boundary of the attainable set. See[2, 21]. It is useful to know that the attainable set has nonempty interior because the maximum principle gives no information otherwise. If we consider nonlinear, autonomous control systems determined on a manifold M by the set ℳk of autonomous, Ck controllable vector fields, then a few technical restrictions on the control system allow us to conclude that there is an open dense subset O of ℳk such that each element of O has an attainable set contained in the closure of its own interior.