Necessary and sufficient condition for global controllability of nonlinear systems

This paper concerns the global controllability of general nonlinear control systems, which is intimately related to their defining vector fields. We give a necessary and sufficient condition by studying how to enlarge the defining set of vector fields without changing its controllability. Four such approaches are obtained. First, for a given set of vector fields, if one of them is reversible, then the images of these given vector fields under the flow of it can be included without changing the controllability. Second, if two of them are reversible, then their Lie bracket can be included to the given set without changing its controllability. Third, the controllability of a given set of vector fields is preserved by positive span. Finally, the controllability of a given set of vector fields is preserved under closure. For a given nonlinear control system, its controllability is determined as follows. Begin with the defining set of vector fields, step by step, we enlarge it by the aforementioned approaches until its controllability can be readily verified. The power of the proposed approaches is demonstrated by examples.