A rest solution, p, of a control system is said to possess controlled stability if for each t1>0 a full nbd. of points of p can be steered to p in time t1 by solutions of the system. Equivalently, if the system acts in a linear space and p is the origin, this states the domain of « null controllability » is open. We consider systems of the form\(X\left( x \right) + \sum\limits_{i = 2}^m {u_i } \left( t \right)Y^i \left( x \right)\) with either each ui measurable with |ui(t)| ⩽1, or |ui| ≤1 inL1 or ‖ui‖ ≤1 in the space of regular countably additive measures, and X, Y2, …, Ym analytic vector fields on an analytic n-manifold. If X(p)=0 a necessary condition that the rest solution p possess controlled stability is that the dimension, at p, of the Lie algebra generated by X, Y2, …, Ym be n. First order sufficient conditions are well known and can be stated in terms of subsets of the elements of this Lie algebra. This paper provides two higher order sufficiency tests, together with examples of their applications.