Geometry of Nonlinear Control with Applications

本课题研究无穷小不可交换流连续族的几何和代数基础。它结合了功能解析算子微积分，即时间演算和代数组合学的方法。这个项目的两个部分的第一部分集中于非线性常微分方程控制的光滑系统。利用新颖的组合结构，如Zinbiel和树形代数，它统一和系统化了许多经典问题的解决技术，包括系统的状态空间实现，路径规划的应用，量子系统的控制，以及科学计算中的几何积分算法。第二部分的动机是来自半导体制造的实际控制问题，旨在将已证明的方法和方法扩展到由双曲守恒定律的非线性系统控制的无限维系统。该项目研究了许多具有实际重要性的动力系统的一个显著特征：不交换的流的共同数学结构。这项研究具有广泛的应用前景。控制设计利用这一特性，通过明智地选择控制动作的顺序，将系统引导到任何期望的状态。在科学计算的分割方法中，碎片缺乏交换性是一个需要处理的复杂问题。对于量子力学系统来说，交换性的缺乏在许多方面是它们本质的本质。除了理论见解、改进的控制设计和高性能计算算法的直接贡献之外，该项目还在水平和垂直上改善了教育和技术基础设施：它将组合学、控制和计算相互联系起来，并将其应用于分子物理学、生物医学和制造系统。它培训研究生，为本科生提供有意义的第一手研究经验，并与现有的倡议合作，培训传统上代表性不足的群体的参与者。

This project investigates the geometric and algebraic foundations of continuous families of infinitesimally noncommuting flows. It combines a functional analytic operator calculus known as chronological calculus together with methods from algebraic combinatorics. The first of two parts of this project focuses on smooth systems governed by nonlinear ordinary differential equations. Utilizing novel combinatorial structures such as Zinbiel and dendriform algebras, it unifies and systematizes solution techniques for a number of classical problems that include state-space realization of systems, applications to path planning, control of quantum systems, and geometric integration algorithms in scientific computing. The second part is motivated by a practical control problem from semiconductor manufacturing and aims to extend proven approaches and methodologies to infinite dimensional systems that are governed by nonlinear systems of hyperbolic conservation laws.This project studies the common mathematical structures underlying a distinguishing feature of many dynamical systems of practical importance: flows that do not commute. This research has broad applications. These include: Control designs exploit this feature to steer systems to any desired states by judiciously choosing the order in which control actions are taken. In splitting methods of scientific computing the lack of commutativity of the pieces is a complication that needs to managed. For quantum mechanical systems the lack of commutativity is in many ways the very essence of their nature. Beyond the immediate contributions of theoretical insights, improved control designs and high performance computing algorithms, this project improves the education and technological infrastructure both horizontally and vertically: It connects combinatorics, control, and computation, with each other and with applications in molecular physics, biomedical and manufacturing systems. It trains graduate students, provide undergraduates with meaningful first-hand research experiences, and it collaborates with existing initiatives to train participants from traditionally underrepresented groups.