Mathematical Sciences: Control Theory and Vector Field Systems

该项目继续对包含自由函数和控制的微分方程式的物理系统建模进行数学研究。分析这类系统的一种方法是使用变换、状态反馈和坐标变化，将系统简化为线性系统。这相当于确定何时可以选择在适当的局部坐标下是线性的矢量场的局部基。虽然这很少被预料到，但下一个最合理的情况是生成幂零或可解李代数的基。该项目将继续为此开展工作。当系统的自由运动或不受控制的部分不存在时，可能会产生迄今最好的结果。否则，主要是由于即使在低维度上也存在大量(不等价的)幂零系统，所以出现了主要障碍。另一个同样重要的问题是确定一个系统的不变量，它将表明是否可以期望相关的幂零或可解李代数。第二条线的调查将集中在被称为联系人地图的某些转换上。这些映射被用来刻画定义在高维欧氏空间上的函数的喷射和喷射丛。微分运算符可以根据喷射和接触变换来定义。它们提供了一种延长解决方案的方法。在这一理论的发展中所面临的两个基本问题是：作为接触变换的射丛自映射的刻画和保持给定流形不变的接触变换的构造。虽然由于Lie和Backlund的原因，有几个与这些问题相关的基本结果，但还需要具体的例子。此外，还需要开发获得接触变形的建设性方法。本研究将遵循的最后一个方向是通过扩展反馈群实现控制系统的线性化，具体地说，将寻找可以通过反馈群获得的控制系统的规范形式。

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.