论文标题
时变矢量字段的指数图
The exponential map for time-varying vector fields
论文作者
论文摘要
表征向量场流量的指数图是理解几何控制理论中控制系统的基本结构属性的关键。但是，由于缺乏通用矢量场的流量缺乏完整性，因此该地图不存在。 为指数图设计了一个适当的替代品，而不是试图强迫流动在歧管上的任何紧凑的假设来全球定义，而是通过对向量场和流动空间的分类发展而定义的，从而允许对此类空间进行系统的定位。也就是说，我们为矢量场的指数图提供了一个可测量的时间依赖性和连续参数依赖性的指数图。此外，从最小的局部Lipschitz依赖到全态和真实的分析依赖性，都认为所有规律性的规律都被考虑。使用适合矢量场和局部差异性的合适拓扑的几何描述，指数图的同态形态是通过对所有规律性案例的均匀处理得出的。最后，证明了一种新的连续依赖性，即固定时间局部流对参数的固定时间流，该参数在建立指数图的同态性中起着重要作用。
The exponential map that characterises the flows of vector fields is the key in understanding the basic structural attributes of control systems in geometric control theory. However, this map does not exists due to the lack of completeness of flows for general vector fields. An appropriate substitute is devised for the exponential map, not by trying to force flows to be globally defined by any compact assumptions on the manifold, but by categorical development of spaces of vector fields and flows, thus allowing for systematic localisation of such spaces. That is to say, we give a presheaf construction of the exponential map for vector fields with measurable time-dependence and continuous parameter-dependence in the category of general topological spaces. Moreover, all manners of regularity in state are considered, from the minimal locally Lipschitz dependence to holomorphic and real analytic dependence. Using geometric descriptions of suitable topologies for vector fields and for local diffeomorphisms, the homeomorphism of the exponential map is derived by a uniform treatment for all cases of regularities. Finally, a new sort of continuous dependence is proved, that of the fixed time local flow on the parameter which plays an important role in the establishment of the homeomorphism of the exponential map.
