Mathematical Sciences: Control Theory and Vector Field Systems

该项目继续对 用微分方程模拟物理系统 包含自由函数的控件。 一种分析方法 这样的系统是使用变换，状态反馈， 例如，坐标变化，以将系统减少到线性 一个. 这相当于确定何时向量的局部基 在适当的局部坐标系中，可以选择 线型 虽然这是很少可以预期的，其次最 合理的情况下将是基生成幂零或 可解李代数 该项目将继续努力， 这个结局 最好的结果是可能的日期时，自由运动 或者系统的不受控部分不存在。 否则， 主要障碍主要是由于 的（不等价）幂零系统存在，即使在低 尺寸. 另一个同样重要的问题是， 确定系统的不变量，该不变量将显示 相关的幂零或可解李代数。 第二条调查线将集中在某些 变换称为接触映射。 这些地图 用于表征定义在上的函数的射流和射流束 高维欧氏空间 微分算子可以 可以用射流和接触变换来定义。 他们 提供一种延长溶液的方法。 基本问题 在这个理论的发展面临着两个： 接触喷注丛自映射的特征 变换和切触变换的构造 使给定的流形保持不变。 虽然有几 基本结果有关这些问题由于李和 贝克兰德，需要具体的例子。 此外，本发明还提供了一种方法， 用于获得接触变换的构造性方法需要 发展。 本研究将遵循的最终方向是 涉及通过扩展反馈的控制系统线性化 一组，特别是，控制系统的规范形式， 可以通过反馈组获得将寻求。

This project continues mathematical research into the modelling of physical systems by differential equations containing free functions, the controls. One means of analyzing such systems is the use of transforms, state feedback and coordinate change for example, to reduce the system to a linear one. This amounts to determining when a local basis of vector fields can be chosen which, in proper local coordinates, is linear. Although this can rarely be expected, the next most reasonable case would be bases which generate nilpotent or solvable Lie algebras. The project will continue work towards this end. Best results to date are possible when the free motion or uncontrolled part of the system is not present. Otherwise, major obstacles are presented due, primarily, to the large number of (inequivalent) nilpotent systems present even in low dimensions. Another, equally important, question is that of determining invariants of a system which will show whether or not related nilpotent or solvable Lie algebras can be expected. A second line of investigation will focus on certain transformations called contact maps. These are maps which are used to characterize jets and jet bundles of functions defined on higher dimensional Euclidean spaces. Differential operators may be defined in terms of jets and contact transformations. They provide a method for prolonging solutions. The basic problems faced in the development of this theory are two: a characterization of self-maps of jet bundles which are contact transformations and the construction of contact transformations which leave a given manifold invariant. While there are several fundamental results related to these questions due to Lie and Backlund, specific examples are needed. In addition, constructive methods for obtaining contact tranformations need to be developed. A final direction this research will follow concerns control system linearization via an extended feedback group, specifically, canonical forms for control systems which can be obtained via the feedback group will be sought.