Mathematical Sciences: Nonlinear Differential Equations, Vector Field Approximations and Control

本研究的重点是发展一个连贯的程序来确定非线性控制系统的稳定性。理解稳定性很重要；在系统理论和系统实践中，几乎所有问题最终都归结为某种形式的稳定问题。线性系统理论的重要性更多地在于测试线性系统的稳定性的便利性，而不是线性本身。早期的工作集中在短时局部可控性问题上，这是稳定的必要条件。前者是开环控制的稳定，后者是闭环控制的稳定。作为先前工作的结果，人们现在有了确定STLC的计算标准。功已经转向用更简单的非线性系统来近似非线性系统，例如幂零系统和相对于膨胀齐次的系统。这项工作为一个相对全面的非线性稳定性理论提供了希望。在本项目中，引入了一个相对于测量稳定的矢量场的新概念。这个概念与非线性动力学的阴影思想密切相关。已经证明，如果Brockett的局部上条件不满足，则非线性系统对于一个相对于测量稳定的向量场是不可局部稳定的。本文还利用齐次逼近和齐次拉格朗日算子，利用近似反馈线性化和最优控制，构造了局部不满足Brockett条件的系统的动态渐近稳定反馈。长期以来，数学力量与工程应用的结合使控制理论成为新数学主题最健康的来源之一。数学家和工程师之间建立的桥梁提供了动态的交流，导致两个领域的进步。这个项目处理的问题是，当稳定状态反馈控制的一些传统测试不存在时，人们可以做些什么。在许多重要问题中，包括具有非完整约束的机械系统和无重力环境下的倒立摆，都需要努力构造不连续状态反馈控制。

The focus of this research is the development of a coherent program to determine the stabilizability of nonlinear control systems. Understanding stabilizability is important; almost all issues in both systems theory and systems practice eventually reduce to some version of the stabilization problem. The importance of linear systems theory rests more on the ease of testing a linear systems for stabilization than on its linearity per se. Earlier work concentrated on the question of short time local controllability, a necessary condition for stabilizability. viewed as stabilization by open loop controls while the latter is stabilization by closed loop controls. As a result of the prior work, one now has computational criteria for deciding STLC. Work has turned to approximations of nonlinear systems by simpler ones such as nilpotent systems and systems which are homogeneous with respect to a dilation. This work holds promise for a relatively comprehensive theory of nonlinear stabilizability. In the present project, a new concept of vector fields which are stable with respect to measurement are introduced. The concept is closely related to the shadowing ideas of nonlinear dynamics. It has been shown already that if Brockett's locally onto conditions not satisfied then a nonlinear system is not locally stabilizable to a vector field which is stable with respect to measurement. Work will also be done using homogeneous approximations and homogeneous Lagrangians to construct dynamic asymptotically stabilizing feedbacks for systems which don't satisfy Brockett's locally onto condition using approximate feedback linearization and optimal control. The combination of mathematical power and engineering applications has long made control theory one of the healthiest sources of new mathematical themes. The bridges built between mathematicians and engineers have provided for dynamic interchanges leading to the advancement of both fields. This project takes up the problem of what one can do when some of the traditional tests for stabilizing state feedback control are not present. In many important problems, including mechanical systems with nonholonomic constraints and the inverted pendulum in a gravity free environment, efforts are to be made to construct discontinuous state feedback controls.