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

本课题研究了非线性系统小时可达集的定性结构和最优控制的分段正则性。对于单输入系统，众所周知，如果系统的尺寸变得太大，最优轨迹可能会表现出不期望的特征，例如颤振的砰砰弧线。这可能是由于缺乏通过控制矢量场影响系统的可能性。为了探索这种情况，本项目试图分析n维的小时间可达集的局部结构，以增加n的值，特别是开发对bang和奇异弧的复杂串联的最优性的测试，ii)研究使用高阶近似锥的多输入系统，该系统考虑了所考虑的轨迹的特定结构。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的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可能的未来涉及非线性组件的设计方法也可能具有实际意义。

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.