Theory and Applications of Finite-Dimensional Nonlinear Control

9803411SussmannResearch将在非线性控制理论上进行，继续首席研究员以前在这一领域的工作，涉及广泛的理论和应用控制理论问题。 所使用的方法将是微分几何控制理论，非光滑分析，和理论的真实的解析映射及其相关的分层。 具体来说，将努力解决一些开放的问题，在最优控制，可控性和实现理论的领域，同时追求必要的数学工具的发展。 特别是，工作将继续在1992年开始的一个重大项目，并已演变自那时以来，并导致了一个强大的，一般的，统一的版本的必要条件的最优性通常被称为``庞特里亚金最大原则。这个最大值原理的新版本需要使用一种新的广义微分理论，称为“多重微分”，部分工作将涉及这一理论及其应用的系统发展。非线性控制的最新发展导致了许多应用于机器人和非完整运动规划的各种问题。 这里的一般问题是为给定的系统找到一条从给定状态到另一个期望状态的路径，满足某些约束或优化某些成本泛函。(For例如，将车辆从给定位置转向另一期望位置，同时避开某些障碍物，可能具有在最短时间内或以最小能量消耗完成的额外要求。 由于最近在计算能力方面的非凡进步，现在期望在真实的时间内解决许多这些问题已经变得现实，我们打算开发有助于这一奋进的方法，通过提供对解决方案结构的先验理解，为了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和一般控制领域的专家之间的交流。数学和工程社区，通过引起社区的注意，应用程序的例子，显示的优势和非线性控制技术的力量。

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.