A Lie-bracket-based notion of stabilizing feedback in optimal control

For a control system two major objectives 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 bound of the functional. This paper is devoted to a crucial step in the programme 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 $ \leq k $ <= 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 as, for instance, a Lyapunov-type inverse theorem - i.e. the possibility of deriving existence of a Minimum Restraint Function from stabilizability - are proved in companion papers.