Mathematical Sciences: Control Theory and Vector Field Systems

该项目继续通过包含自由函数和控制的微分方程对物理系统建模进行数学研究。分析这种系统的一种方法是使用变换，例如状态反馈和坐标变化，将系统简化为线性系统。这相当于确定什么时候可以选择向量场的局部基，它在适当的局部坐标下是线性的。虽然这种情况很少发生，但下一个最合理的情况是产生幂零或可解李代数的基。该项目将继续为此目的而努力。当系统的自由运动或不受控制的部分不存在时，迄今为止最好的结果是可能的。否则，主要的障碍是由于即使在低维中也存在大量的（不等价的）幂零系统。另一个同样重要的问题是确定一个系统的不变量，它将表明相关的幂零李代数或可解李代数是否可以预期。第二项调查将集中在称为接触图的某些转换上。这些映射用来描述在高维欧几里德空间中定义的函数的射流和射流束。微分算子可以用射流和接触变换来定义。它们提供了一种延长解决方案的方法。发展这一理论所面临的基本问题有两个：射流束的自映射的表征，即接触变换和接触变换的构造，接触变换使给定的流形不变。虽然由于Lie和Backlund，有几个与这些问题相关的基本结果，但需要具体的例子。此外，需要发展获得接触变换的建设性方法。本研究将遵循的最后一个方向是通过扩展反馈群来实现控制系统的线性化，具体而言，将寻求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.