One of the fundamental problems in control theory is that of controllability. Indeed, many design methodologies rely on some hypotheses that concern controllability. The problem of controllability is essentially one of describing the nature of the set of states reachable from an initial state. In the development of this theory, there are two properties that arise as being important. The first is the property of “accessibility,” which means that the reachable set has a nonempty interior. The treatment of accessibility we present follows the approach of the fundamental paper of Sussmann and Jurdjevic . Results of a related nature are those of Krener  and Hermann and Krener . The property of “controllability” extends accessibility by further asking whether the initial state lies in the interior of the reachable set. The matter of providing general conditions for determining controllability is currently unresolved, although there have been many deep and insightful contributions. While we cannot hope to provide anything close to a complete overview of the literature, we will mention some work that is commensurate with the approach that we take here. Sussmann has made various important contributions to controllability, starting with the paper [Sussmann 1978]. In the paper [Sussmann 1983], a Lie series approach was developed for the controllability of control-affine systems, and this approach culminated in the quite general results of [Sussmann 1987], which incorporated the ideas of Crouch and Byrnes  concerning input symmetries.
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