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.

中文翻译：

---

非线性系统全局可控的充要条件

本文关注一般非线性控制系统的全局可控性，这与它们定义的矢量场密切相关。我们通过研究如何在不改变其可控性的情况下扩大向量场的定义集，给出了一个充要条件。获得了四种这样的方法。首先，对于给定的一组向量场，如果其中一个是可逆的，那么在不改变可控性的情况下，这些给定向量场在它的流动下的图像就可以被包含进来。其次，如果它们中的两个是可逆的，那么它们的李括号可以包含在给定的集合中，而不会改变其可控性。第三，给定向量场集的可控性由正跨度保持。最后，给定向量场集的可控性在闭包下得以保留。对于给定的非线性控制系统，其可控性确定如下。从定义向量场集开始，我们逐步通过上述方法将其放大，直到可以轻松验证其可控性。所提出的方法的威力通过例子来证明。