Mathematical Sciences:Theory and Applications of Finite- Dimensional Nonlinear Control

9500798 Sussmann研究将在非线性控制理论上进行，继续主要研究者以前在这一领域的工作，研究最优控制、机器人应用、反馈稳定和非完整运动规划等广泛的问题。 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。 必要的数学 将开发工具来解决最优控制、可控性和实现理论领域中的一些开放性问题。 特别是，最近的主要研究结果的庞特里亚金最大值原理系统的向量场将得到扩展，其目标是证明一个一般的最大值原理系统的微分包含在最小的假设。 最佳轨迹的规律性问题也将得到解决。 所提出的研究符合控制理论的一般框架，该框架涉及设计问题，其中动态系统的某些组件可以访问控制代理，该控制代理可以修改它们以改变系统的行为。 其目的是控制系统以实现期望的行为或满足某些规范。 这种情况在工程学以及所有的物理学和生物学中到处都有，而且从数学的观点来看，许多表面上不同的问题原来具有共同的结构特征。 控制系统的数学理论的目的是以系统的方式研究这些特征，并开发解决特定设计问题的技术，例如使机器人手臂执行特定任务，或保持天线聚焦。 在这个项目中考虑的问题处理（a）最优控制，其中的目标是确定如何控制系统，以优化某些标准;（B） 稳定性，目标是将系统驱动到某个期望的操作点，并使其保持在那里;（c）运动规划，试图找到一种控制策略，使系统沿着某个规定的路径运动。*

9500798 Sussmann Research will be carried out on nonlinear control theory, continuing the principal investigator s previous work in this area on a broad class of problems in optimal control, applications to robotics, feedback stabilization, and nonholonomic motion planning. The methods used will be those of differential geometric control theory, nonsmooth analysis, and the theory of real analytic maps and their associated stratifications. Necessary mathematical tools will be developed to solve a number of open problems in the areas of optimal control, controllability, and realization theory. In particular, recent results of the principal investigator on the Pontrayagin Maximum Principle for systems of vector fields will be extended, with the goal of proving a general Maximum Principle for systems of differential inclusions under minimal hypotheses. The question of the regularity of optimal trajectories will also be addressed. The proposed research fits within the general framework of control theory, which is concerned with design problems in which some components of a dynamical system are accessible to a controlling agent that can modify them so as to alter the system s behavior. The object is to control the system so as to achieve a desired behavior, or meet some specifications. Situations of this kind occur everywhere in engineering as well as in all of the physical and biological sciences, and many apparently different problems turn out to have common structural features from the mathematical point of view. The purpose of the mathematical theory of control systems is to study these features in a systematic way and to develop techniques for solving particular design problems, such as causing a robot arm to carry out a particular task, or keeping an antenna focused. The issues considered in this project deal with (a) optimal control, in which the goal is to determine how to control a system so as to optimize some criterion; (b) stabiliz ation, where the object is to drive the system towards some desired operating point and keep it there; and (c) motion planning, where it is sought to find a control strategy that will cause a system to move along some prescribed path. ***