Mathematical Sciences: The Geometry of Highly Nonlinear Control Systems

该研究计划旨在开发一种 几何基础的综合理论， 高度非线性控制系统的动态特性， 很难用线性近似来描述。 是 旨在首先关注反馈的实际目的 高度非线性系统的稳定化。 特别 直接利益是经典零的推广， 动力学流形仅由一些高阶条件（立即 与通过奇点的跟踪有关）。 这是设想中的 为了使可控性条件的发展并行， 从线性，到一阶非线性，最后到高阶 非线性条件 第二个主要方向涉及 自由李代数元素工具的开发 (and他们满足的关系），使一个简洁地 描述稳定性的条件，并描述 可控性和最优性。 最重要的是 滤子与自由幂零基的特征 李代数（在有限数量的生成元上）， 对于所考虑的问题。 这项工作是一个延续 研究计划已经从可控性扩展到 的几何和拓扑特性的可视化 控制系统

This research program is aimed at developing a comprehensive theory of the geometrical foundations that underlie the dynamics of highly nonlinear control system, that can only very poorly be described by linear approximations. It is intended to focus first on the practical purposes of feedback stabilization of highly-nonlinear systems. Of particular immediate interest is a generalization of the classical zero- dynamics manifold only by some high-order condition (immediately related to tracking through singularities). This is envisioned to parallel the development of condition for controllability, from linear, over first order nonlinear to eventually high-order nonlinear conditions. A second main direction involves the development of tools in terms of elements of a free Lie algebra (and the relations they satisfy) that allow one to concisely describe conditions for stabilizability, and also characterize controllability and optimality. Most important are the characterization of filtrations and of bases of a free nilpotent Lie algebra (on a finite number of generators) that are suitable for the problems considered. This work is a continuation of a research program that has expanded from controllability, to the visualization of geometrical and topological properties of the control system.