This CAREER proposal proposes a fundamental investigation of a new class of control problems, Ensemble Control, which involves controlling a large number of dynamical systems with the same control signal. State-of-the-art quantum technology can trap and experiment with individual atoms, image brains as well as generate structural and dynamical information of biological macromolecules. Numerous applications arising from such emerging techniques involve controlling a large quantum ensemble, e.g., on the order of Avogadro number (6X1023), by use of the same control field. In many cases, the elements of the ensemble show variations in the values of the parameters characterizing the system dynamics. For example, in magnetic resonance experiments, nuclear spins of an ensemble may have a dispersion in their natural frequencies, so that the spins of an ensemble may have a dispersion in their natural frequencies, so that spins with different excitations (pulse sequences) that can steer such an ensemble of systems with different dynamics from an initial state of desired final state. The challenge is to simultaneously steer a continuum of systems between points of interest with the same control signal. We plan to provide controllability conditions and optimal control techniques for such under-actuated systems. The resulting methodology, via a systematic study of ensemble control problems, can be directly applied to the design of optimal pulse sequences in Nuclear Magnetic Resonance (NMR) spectroscopy and imaging (MRI).
The proposed work will advance stat-of ?the-art methods of mathematical control theory and has broad applications including medical imaging, structural biology, as well as quantum information processing. The PI proposes not only to make best use of engineering expertise in above areas of biology and quantum mechanics, but also to help make methods and problems in these fields more accessible to engineers. The proposed CAREER plan provides educational activities and research opportunities for both undergraduates and graduate students to work on multidisciplinary projects. They will work in close collaboration with scientists in other disciplines and will work on solving real world problems. For example, students will design optimal pulse sequences for MRI systems and will implement them onto the real MRI machines.
The objective of this project is a fundamental investigation of ensemble control problems involving the guidance of a large number or a continuum of structurally identical dynamical systems, with slightly different dynamics, by using a common control input. This class of problems originates from practical pulse sequence design in nuclear magnetic resonance (NMR) spectroscopy and imaging (MRI), and is ubiquitous in broad areas of science and engineering, such as quantum physics, neuroscience, and robotics. Intellectual Merit: The achievements made with this funding support lead to novel theoretical and practical developments in systems and control theory. Controllability of general linear and some forms of bilinear and nonlinear ensemble systems has been investigated through the development of new methods that involve translating the controllability analysis to a problem of polynomial approximation for systems evolving on the special orthogonal group or to the solvability of a Fredholm integral equation for linear ensemble systems. Computational methods have also been developed for solving optimal ensemble control problems. The robustness of such methods has been numerically tested and experimentally verified through designing optimal pulses for protein NMR spectroscopy and optimal waveforms for synchronization and entrainment of weakly forced nonlinear chemical oscillators. The scope and impact of this work extends beyond the control of quantum systems. The resulting analytical and numerical methods have been effectively applied to the optimal manipulation and synchronization of neural, chemical, and uncertain engineering systems, which in turn broaden seminal applications in neuroscience, electrochemistry, and robotics. Broader Impact: The proposed activities promoted research and education through interdisciplinary investigation in engineering, physics, chemistry, and neuroscience. The outcomes have been disseminated broadly to these research communities through publications, presentations at conferences and universities, and distributed software packages. The PI has held a joint appointment in the Division of Biology & Biomedical Sciences (DBBS) at Washington University in St. Louis and has advised students across disciplines from electrical and systems engineering, biomedical engineering, physics, and mathematics. The PI has also been involved in Annual Junior Scientist Institute organized by DBBS to help train 6-8th graders who are highly interested in the sciences and engaged with Science Outreach Programs at Washington University. The conducted multidisciplinary research through this career award has been published, during the funding support period, in eighteen papers in leading scientific journals in disciplines including control theory and dynamical systems, applied mathematics, physics, and bioengineering, and twenty papers in peer-reviewed proceedings of international conferences.
