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.