Mathematical Sciences: Geometric and Computational Foundations of Nonlinear Feedback Control

正在进行的空气和航天器控制工作， 非线性动力学提出了许多新的几何问题， 控制理论 在以前的工作中，PI介绍了 柯西特征的多叶结构，与一个 控制系统 这是一个基本的几何不变的 反馈结构，最终决定（本地和 全局上）通过应用状态反馈可以实现什么 控制设备。 这项拟议的工作将继续进行， 对柯西特征结构的研究 传统数学的观点（例如， cauchy特征结构确定控制系统 反馈等效？） 从LISP/Symbolic计算 的观点 与此同时，工作也将继续在谎言- 各种数值算法的理论基础， 非线性微分方程。 例如， 经典Runge-Kutta算法的思想被实现为 可微流形上的微分方程 带有李群结构的 许多潜在的应用 例如，对于旋转和非线性的控制， 机械系统，如在飞机研究中遇到的， 机器人 该项目属于控制科学的广泛领域福尔斯， 这是一个跨学科的研究活动， 数学和工程学

Work underway on Control of air and spacecraft that have nonlinear dynamics suggests many new problems of geometric control theory. In previous work, the PI has introduced the cauchy characteristic multifoliate structure associated with a control system. It is a basic geometric invariant of the feedback structure, ultimately determining (both locally and globally) what can be accomplished by applying state-feedback control to the apparatus. This proposed work will carry on the study of the cauchy characteristic structure both from the point of view of traditional mathematics (e.g. to what extent does the cauchy characteristic structure determine the control system up to feedback equivalent?) and from the LISP/Symbolic computation point of view. In parallel, work will also continue on the Lie- theoretic foundation of various numerical algorithms for dealing with nonlinear differential equations. For example, how can the idea of the classical Runge-Kutta algorithm be implemented for differential equations which live on a differentiable manifold that carries a Lie group structure? Many potential applications are possible e.g. to the control of rotating and nonlinear mechanical systems, as encountered in the study of aircraft and Robotics. This project falls into the broad area of control science, which is an interdisciplinary research activity involving both mathematics and engineering.