时变向量场的指数映射
The exponential map for time-varying vector fields
摘要：
在几何控制理论中，描述向量场流的指数映射是理解控制系统基本结构属性的关键。然而，由于一般向量场的流缺乏完整性，该映射不存在。一个适当的替代品的指数映射，而不是试图迫使流动被全球定义的任何紧凑的假设上的歧管，但分类发展的空间的向量场和流量，从而允许系统本地化的空间。也就是说，在一般拓扑空间范畴中，我们给出了具有可测时间依赖和连续参数依赖的向量场的指数映射的预层构造。此外，考虑了状态的所有正则性，从最小局部Lipschitz依赖到全纯和实解析依赖。利用向量场和局部同胚的适当拓扑的几何描述，通过对所有情形的同胚的统一处理，导出了指数映射的同胚。最后，证明了一类新的连续依赖性，即定时局部流对参数的依赖性，该参数在指数映射同胚的建立中起着重要作用。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
