Chronological Calculus and Nonlinear Control

时序演算（CC）是计算和分析时变非线性矢量场和非交换流的一种工具。它是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。本项目将使用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。典型的问题是Zinbiel代数中的最优控制曲率和理想结构。一个挑战是将组合学和非结合代数中的形式化工作与这些形式化对象如何映射到解析和几何结构的严格证明合并在一起。一个关键的目标是了解底层几何如何影响和限制动态系统的结构行为，模拟不同的应用。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，使它们能够进行高性能计算。该项目将在横向和纵向上产生更广泛的影响：要开发的工具和方法可立即应用于各种学科，基本上适用于任何需要采取行动顺序的地方。在这个项目中考虑的系统类别包括量子系统、高性能数值计算、运筹学和生物医学科学的许多领域，从种群动力学到药物动力学和分子生物学。该项目以应用问题为动力，开发新的数学工具，并将其提供给应用科学，从而加强跨学科的联系。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整合和当地基础设施，该课程作为管道，将来自不同背景的新人才聚集在一起，成为发现过程的积极参与者。主要手段是将控制理论和生物科学联系起来的令人兴奋的问题，以及在这个项目中使用交互式可视化工具的高度实验性。这两种方法都可以让本科生以最低的正式先决条件开始有意义的参与，同时无缝地进入高级理论工作。

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.