Geometry of Nonlinear Control with Applications

非线性控制几何及其应用

• 批准号: 0908204
• 负责人: Matthias Kawski
• 金额: $ 20万
• 依托单位: Arizona State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0908204&HistoricalAwards=false
• 关键词: Geometry; Nonlinear; Control; Applications

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.

本计画研究无穷小非交换流之连续族之几何与代数基础。 它结合了一个功能分析算子演算称为时序演算与方法从代数组合。 这个项目的两个部分中的第一部分集中在由非线性常微分方程控制的光滑系统上。 利用新的组合结构，如Zinbiel和dendriform代数，它统一和系统化的解决方案技术的一些经典问题，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，和几何积分算法在科学计算。 第二部分的动机是从半导体制造的实际控制问题，并旨在扩展已证明的方法和方法，以无穷维系统，由双曲守恒律的非线性系统。该项目研究的共同数学结构下的一个显着特点，许多动力系统的实际重要性：流不交换。 这项研究具有广泛的应用。其中包括：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。 在科学计算的分裂方法中，缺乏部分的可交换性是一个需要管理的复杂性。 对量子力学系统而言，缺乏交换性在许多方面是其本质。 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外，该项目还在横向和纵向上改善了教育和技术基础设施：它将组合学，控制和计算相互连接起来，并与分子物理，生物医学和制造系统中的应用相结合。 它培训研究生，为本科生提供有意义的第一手研究经验，并与现有的举措合作，培训传统上代表性不足的群体的参与者。