We study a class of systems described by a family of parameterized dynamical equations. Although the form of the differential equation is consistent, the parameters that characterize the motion show variation over the ensemble. From the perspective of control, the problem is to drive this whole family of structurally similar systems from an initial state (distribution) to target state (distribution), compensating for the variation in parameters. The ultimate challenge is that due to constraints on the system, we are only able to inject a single (common to all systems), open-loop (without feedback) input to drive the systems.
Pulse Sequence Design for NMR & MRI
There is a wealth of opportunity to apply optimal control techniques to solve the challenging problems that arise in nuclear magnetic resonance (NMR) and the technologically similar medical imaging modality, MRI. This work has the potential to boost fidelity of NMR spectrum, which is used to determine the structure of biological macromolecules (e.g., in drug design). This same methodology will also enhance the resolution of MRI images which will facilitate better diagnoses, for example, for early stage cancer detection. Some of our work (see our 2011 Proceeding of the National Academy of Sciences paper) has been shown to provide up to increase NMR sensitivity by as much as twenty times over conventional methods.
Although each atom in the periodic table (e.g. a hydrogen atom) has a characteristic frequency in a given magnetic field, due to environmental and chemical effects as well as experimental imperfection this frequency instead shows variation around the nominal value. During the imaging process, the atoms (spins with magnetic moments) are manipulated using electromagnetic waves (radio-frequency pulses) to align them in different directions. Due to the variation in frequency (as well as variation in other parameters) the naive inputs based on the nominal frequency values cause the systems to scatter rather than stay together. This scatter, or dispersion, directly leads to loss of signal strength. In this work, we aim to design radio-frequency pulses (inputs) that compensate for the inherent variation in parameters by casting pulse sequence design problems as optimal ensemble control problems.