Theory and Applications of Finite-Dimensional Nonlinear Control

9803411 sussmann将对非线性控制理论进行研究，继续首席研究员以前在这一领域的广泛的理论和应用控制理论问题的工作。使用的方法将是微分几何控制理论、非光滑分析和真实分析图及其相关分层理论。具体而言，将努力解决最优控制，可控性和实现理论领域的一些开放问题，同时追求必要的数学工具的发展。特别是，从1992年开始的一个重大项目的工作将继续下去，从那时起，这个项目不断发展，并导致了一个强大的、通用的、统一的最优性必要条件的版本，通常被称为“庞特里亚金最大原理”。这个新版本的极大原理需要使用一种新的广义微分理论，称为“多重微分”，部分工作将涉及系统地发展这一理论及其应用。非线性控制在机器人和非完整运动规划方面的最新发展已经导致了许多应用。这里的一般问题是为一个给定的系统找到一条路径，使它从一个给定的状态到达另一个期望的状态，满足一些约束或优化一些成本函数。（例如，在避开某些障碍物的同时，将车辆从给定位置驾驶到另一个期望位置，这可能需要在最短的时间内或以最小的能量消耗完成。）由于最近计算能力的非凡进步，现在期望实时解决许多这些问题已经成为现实，我们打算开发有助于这一努力的方法，通过提供对解决方案结构的先验理解，从而减少寻找解决方案所需的搜索。将作出特别的努力，以改善非线性控制领域的专家与一般数学和工程界之间的交流，通过提请社区注意显示非线性控制技术的优点和力量的应用实例。

9803411SussmannResearch will be carried out on nonlinear control theory, continuing the principal investigator's previous work in this area on a broad class of theoretical and applied control theory problems. The methods used will be those of differential-geometric control theory, nonsmooth analysis, and the theory of real analytic maps and their associated stratifications. Specifically, efforts will be made to solve a number of open problems in the areas of optimal control, controllability, and realization theory, while pursuing the development of the necessary mathematical tools. In particular, work will continue on a major project that began in 1992, and has evolved since then and has led to a strong, general, unified version of the necessary conditions for optimality usually known as the ``Pontryagin Maximum Principle.'' This new version of the Maximum Principle requires the use of a new theory of generalized differentials, called ``multidifferentials,'' and part of the work will involve the systematic development of this theory and its applications.Recent developments in nonlinear control have led to many applications to various issues in robotics and nonholonomic motion planning. The general question here is that of finding a path for a given system that takes it from a given state to another desired state, satisfying some constraints or optimizing some cost functional. (For example, steer a vehicle from a given position to another desired position while avoiding certain obstacles, possibly with the extra requirement that this be done in minimum time or with minimum expenditure of energy.) Thanks to the extraordinary recent advances in computing power, it has now become realistic to expect to solve many of these problems in real time, and we intend to develop methods that will contribute to this endeavor, by providing an a priori understanding of the structure of the solutions, so as to reduce the search needed to find it. A special effort will be made to improve communication between specialists in the field of nonlinear control and the general mathematical and engineering community, by bringing to the attention of the community examples of applications that show the advantages and the power of the techniques of nonlinear control.