Mathematical Sciences: The Structure of the Small-Time Reachable Set and Regularity Properties of Optimal Trajectories for Control- Linear Systems

数学科学：控制线性系统的小时间可达集的结构和最优轨迹的正则性质

• 批准号: 8820413
• 负责人: Heinz Schaettler
• 金额: $ 4.43万
• 依托单位: Washington University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-01 至 1991-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8820413&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Structure; Small; Time

This project addresses the qualitative structure of small-time reachable sets and piecewise regularity properties of optimal controls for nonlinear systems. For single-input systems it is known that optimal trajectories can exhibit undesirable features such as chattering bang-bang arcs if the dimension of the system becomes too large. This may be due to a lack of possibility to influence the system via the control vector field. To explore that scenario this project seeks to analyze the local structure of small-time reachable sets in dimension n for increasing values of n, in particular, to develop tests for optimality of complex concatenations of bang and singular arcs, and ii) to investigate multi-input systems using higher-order approximating cones which take into account specific structures of trajectories under consideration. The underlying technical framework is provided by differential-geometric descriptions and Lie-algebraic computational techniques. The long term aim of this work is to achieve a qualitative understanding of the local behavior of nonlinear control systems. Aside from its evident theoretical relevance, such a knowledge might also be practically of interest for any possible future approach to design involving nonlinear components.

该项目解决了小时间可达集的定性结构和非线性系统最优控制的分段正则性质。 对于单输入系统，众所周知，如果系统尺寸变得太大，最佳轨迹可能会表现出不良特征，例如颤动的爆炸弧。 这可能是由于缺乏通过控制矢量场影响系统的可能性。 为了探索该场景，该项目旨在分析 n 维中小时间可达集的局部结构，以增加 n 的值，特别是开发爆炸和奇异弧的复杂串联的最优性测试，以及 ii）使用考虑到所考虑的轨迹的特定结构的高阶近似锥来研究多输入系统。 底层技术框架由微分几何描述和李代数计算技术提供。 这项工作的长期目标是实现对非线性控制系统局部行为的定性理解。 除了其明显的理论相关性之外，这样的知识对于任何未来可能涉及非线性组件的设计方法也可能具有实际意义。