Award Abstract #0509030
Chronological Calculus and Nonlinear Control

NSF Org:	DMS Division of Mathematical Sciences
Initial Amendment Date:	July 21, 2005
Latest Amendment Date:	July 21, 2005
Award Number:	0509030
Award Instrument:	Standard Grant
Program Manager:	Mary Ann Horn DMS Division of Mathematical Sciences MPS Directorate for Mathematical & Physical Sciences
Start Date:	August 1, 2005
Expires:	July 31, 2009 (Estimated)
Awarded Amount to Date:	$180000
Investigator(s):	Matthias Kawski kawski@asu.edu (Principal Investigator)
Sponsor:	Arizona State University ORSPA TEMPE, AZ 85287 480/965-5479
NSF Program(s):	APPLIED MATHEMATICS
Field Application(s):	0000099 Other Applications NEC
Program Reference Code(s):	OTHR, 0000
Program Element Code(s):	1266

ABSTRACT

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.

PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH

Matthias Kawski. "On the problem whether controllability is finitely determined," Proceeding of MTNS 2006 (17th Mathematical Theory of Networks and Systems), Kyoto, Japan)., 2006, p. n/a.

Matthias Kawski. "On the problem whether controllability is finitely determined," Proceeding of MTNS 2006 (17th Mathematical Theory of, 2006, p. n/a.

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