  For a control system two major issues can be considered: the stabilizability
with respect to a given target, and the minimization of an integral functional
(while the trajectories reach this target). Here we consider a problem where
stabilizability or controllability are investigated together with the further
aim of a "cost regulation", namely a state-dependent upper bounding of the
functional. This paper is devoted to a crucial step in the program of
establishing a chain of equivalences among degree-k stabilizability with
regulated cost, asymptotic controllability with regulated cost, and the
existence of a degree-k Minimum Restraint Function (which is a special kind of
Control Lyapunov Function). Besides the presence of a cost we allow the
stabilizing "feedback" to give rise to directions that range in the union of
original directions and the family of iterated Lie bracket of length less or
equal to $k$. In the main result asymptotic controllability [resp. with
regulated cost] is proved to be necessary for degree-k stabilizability [resp.
with regulated cost]. Further steps of the above-mentioned logical chain are
proved in companion papers, so that also a Lyapunov-type inverse theorem --
i.e. the possibility of deriving existence of a Minimum Restraint Function from
stabilizability -- appears as quite likely.
