Mathematical Sciences: Geometric and Computational Foundations of Nonlinear Feedback Control

具有非线性动力学的空气和航天器的控制工作提出了几何控制理论的许多新问题。在以前的工作中，PI引入了与控制系统相关的柯西特征多叶结构。它是反馈结构的基本几何不变量，最终决定（局部和全局）通过对设备应用状态反馈控制可以完成什么。这项工作将从传统数学的角度（例如，柯西特征结构在多大程度上决定控制系统达到反馈等效？）和从LISP/符号计算的角度对柯西特征结构进行研究。同时，还将继续研究处理非线性微分方程的各种数值算法的李论基础。例如，如何将经典龙格-库塔算法的思想应用于带有李群结构的可微流形上的微分方程？许多潜在的应用是可能的，例如旋转和非线性机械系统的控制，如在飞机和机器人的研究中遇到的。本项目属于控制科学的广泛领域，是一项涉及数学和工程的跨学科研究活动。

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.