李群和齐性空间上控制系统与线性系统的等价性
Equivalence of Control Systems with Linear Systems on Lie Groups and Homogeneous Spaces
摘要：
本文的目的是证明流形上的控制仿射系统通过同构等价于李群或齐性空间上的线性系当且仅当该系统的向量场是完备的并生成有限维李代数。连通李群上的向量场是线性的，如果它的流是一个单参数自同构自同构群。通过增加一个左不变向量场得到一个仿射向量场。它在齐性空间上的投影，无论何时存在，仍称为仿射。齐性空间上的仿射向量场可以用右不变向量场的投影的李括号来刻画。齐性空间上的线性系是漂移部分是仿射的，而其控制部分是不变的。主要结果是基于一般定理有限维代数所产生的完全向量场，密切相关的定理宫，并有自己的利益。本证明利用几何控制理论的论点。
英文摘要：
The aim of this paper is to prove that a control affine system on a manifold is equivalent by diffeomorphism to a linear system on a Lie group or a homogeneous space if and only the vector fields of the system are complete and generate a finite dimensional Lie algebra. A vector field on a connected Lie group is linear if its flow is a one parameter group of automorphisms. An affine vector field is obtained by adding a left invariant one. Its projection on a homogeneous space, whenever it exists, is still called affine. Affine vector fields on homogeneous spaces can be characterized by their Lie brackets with the projections of right invariant vector fields. A linear system on a homogeneous space is a system whose drift part is affine and whose controlled part is invariant. The main result is based on a general theorem on finite dimensional algebras generated by complete vector fields, closely related to a theorem of Palais, and which have its own interest. The present proof makes use of geometric control theory arguments.
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