Geometry of Nonlinear Control with Applications

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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.