Theory and Applications of Finite-Dimensional Nonlinear Control

苏斯曼将对非线性控制理论进行研究，继续主要研究人员在这一领域广泛的理论和应用控制理论问题上的工作。所使用的方法将是微分几何控制论、非光滑分析和真实解析映射及其相关分层的理论。具体地说，将努力解决最优控制、可控性和实现理论领域的一些未决问题，同时追求必要的数学工具的发展。特别是，将继续开展1992年开始的一个重大项目的工作，该项目自那时以来一直在发展，并产生了通常称为“庞特里亚金最大值原则”的最优必要条件的强有力、一般性和统一化版本。这一新版本的最大值原理需要使用一种新的广义微分理论，称为“多重微分”，部分工作将涉及该理论及其应用的系统发展。非线性控制的最新发展已导致许多应用于机器人学和非完整运动规划中的各种问题。这里的一般问题是为给定的系统找到一条路径，使其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，将车辆从一个给定的位置转向另一个所需的位置，可能需要在最短的时间或最小的能量消耗下完成这一额外要求。)由于最近计算能力的非凡进步，现在期望实时解决其中许多问题已经变得现实，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了解，以减少寻找它所需的搜索。将特别努力改善非线性控制领域的专家与一般数学和工程界之间的沟通，提请社区注意显示非线性控制技术的优势和力量的应用实例。

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.