Chronological Calculus and Nonlinear Control

时序演算(CC)是一种计算和分析时变非线性矢量场和非通勤流的工具。它基于将动力系统提升到光滑函数代数上的线性算子空间的经典技术。这个项目将使用CC来研究完全非线性控制系统的几何和代数基础，因为它们通常出现在生物医学科学的模型中。典型的项是最优控制的曲率和Zinbiel代数中的理想结构。一个挑战是将组合学和非结合代数中的形式工作与这些形式对象如何映射到解析和几何结构的严格理由结合在一起。一个关键的目标是了解底层几何结构如何影响和限制模拟不同应用的动力系统的结构行为。此外，在这个项目中开发的代数和组合工具也产生了紧凑的算法工具，有助于高性能计算。这个项目将产生不同的更广泛的影响，无论是水平还是垂直：要开发的工具和方法立即适用于不同的学科集，基本上在任何重要的地方，采取行动的顺序是重要的。本项目中考虑的系统类别的用途包括量子系统、高性能数值计算、运筹学，以及生物医学科学中的许多领域，从种群动力学到药物动力学和分子生物学。在应用问题的推动下，该项目开发了新的数学工具，并使其可用于应用科学，从而加强了跨学科联系。作为横向影响的补充，该项目还将通过开发有吸引力的控制课程来加强垂直一体化和当地基础设施，作为管道，将来自所有背景的新人才输送到发现过程的积极参与者中。主要手段是连接控制理论和生物科学的激动人心的问题，以及在这个项目中使用交互式可视化工具的高度实验性。这两者都允许本科生开始有意义的参与，只需最少的正式先决条件，同时无缝地导致高级理论工作。

The chronological calculus (CC) is a tool for computation and analysis of time-varying nonlinear vector fields and noncommuting flows. It is based on the classical technique of lifting dynamical systems to the space of linear operators on an algebra of smooth functions. This project will use the CC to investigate the geometric and algebraic foundations of fully nonlinear control systems as they commonly appear, in particular, in models in the bio-medical sciences. Typical items are the curvature of optimal control, and ideal structures in Zinbiel algebras. One challenge is to merge the formal work in combinatorics and nonassociative algebra with rigorous justifications how these formal objects map onto analytic and geometric structures. A key objective is to understand how the underlying geometry affects and limits the structural behavior of dynamical systems that model diverse applications. Moreover, the algebraic and combinatorial tools developed in this project also yield compact algorithmic tools that lend themselves to high performance computation.This project will have diverse broader impacts, both horizontally and vertically: The tools and methodologies to be developed are immediately applicable in a diverse set of disciplines, basically everywhere where it matters in which order actions are taken. The classes of systems considered in this project have uses that include quantum systems, high performance numerical computation, operations research, and many areas in the biomedical sciences, from population dynamics to pharmacokinetics and molecular biology. Motivated by applied problems, this project develops new mathematical tools, and makes them available to the applied sciences, thereby strengthening interdisciplinary ties. Complementing the horizontal impacts, this project will also enhance the vertical integration and the local infrastructure by developing an attractive control curriculum that acts as a pipeline to funnel new talents from all backgrounds into becoming active participants in the discovery process. Principal means for this are exciting problems that connect control theory and the biosciences, and the highly experimental nature of using interactive visualization tools in this project. Both of these allow undergraduates to start meaningful participation with only minimal formal prerequisites while seamlessly leading to advanced theoretical work.