# Project Details

### Description

Intellectual Merit. A distinguishing feature of many dynamical systems of practical importance are ows that do not commute. This means that the order in which they are followed matters. In the context of controlled dynamical systems this feature is exploited to steer systems to any desired states. The key is to judiciously design the order in which control actions are taken. In various other settings, such as splitting methods in scientic computing, the lack of commutativity of the pieces is less a desirable feature, but rather a nuisance and complication. Yet the mathematical rules, that govern the interactions of the pieces are still the same. This project advances the understanding of continuous families of innitesimally noncommuting ows, and it and develops tools for their analysis and design. It uses a functional analytic operator calculus known as chronological calculus together with combinatorial methods to investigate the geometric and algebraic foundations of nonlinear control theory. The rst part of the project focuses on smooth nonlinear systems governed by ordinary dierential equations. Utilizing recently developed combinatorial tools it unies and systematizes solution techniques for a number of classical problems that include state-space realization of systems given by input-output relations, applications to path planning and control of quantum systems, and geometric integration algorithms. The second part is motivated by a practical control problem from semiconductor manufacturing. One of its objectives is to extend proven approaches and methodologies to innite dimensional systems that are governed by nonlinear hyperbolic systems of conservation laws, a class of partial dierential equations. The broader impacts of this project will be manifold, both horizontally and vertically. The tools and methodologies developed in the rst part are applicable to many disciplines beyond classical control theory, including e.g. numerical computing, molecular physics, and potentially medical imaging. The second part directly addresses problems from semi-conductor manufacturing with potentially very high economic impact. Beyond training graduate students and fully establishing novel courses, the project makes special eorts at providing undergraduates with meaningful rst-hand research experiences. The project reaches out in its technical focus, and will collaborate with existing initiatives to attract to control theory students from traditionally underrepresented groups. The project supports eorts by the PI such as the Preparing Future Faculty Program, and articulating with the large communities focused on undergraduate curricula, especially the use of computing technology in general, and computer algebra systems in particular.

Status | Finished |
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Effective start/end date | 9/15/09 → 8/31/13 |

### Funding

- National Science Foundation (NSF): $200,000.00