Periodic Pulsed Controllability with Applications to NMR

Abstract

In this thesis we study a class of problems that require simultaneously controlling a large number of dynamical systems, with varying system dynamics, using the same control signal. We call such problems ensemble control problems, as the goal is to simultaneously steer the entire ensemble of systems. These problems are motivated by many physical systems and we will be particularly interested in the manipulation of nuclear spins in Nuclear Magnetic Resonance (NMR) experiments. System dispersion arise from imprecise magnets for controls, or from disruptive intermolecular interactions. In all cases, the aim is to attenuate the aspects fo the dynamics that correspond to noise or errors, while perserving the aspects that contain the quantities of interest. In liquid NMR experiments this could correspond to preserving Larmor frequency in the presence of inhomogeneities of the strength of the applied radio frequency (RF) field. In solid state NMR, reducing or eliminating orientation dependent magnetic fields is of key concern, so that a precise spectrum can be observed. We approach the problem from the standpoint of mathematical control theory in which the challenge is to simultaneously steer a continuum of systems between points of interest with the same control signal. At the heart of this problem is finding ways for the nonlinearity of the system to be used to our advantage, so that while all members of the ensemble will be driven with the same controls, their final orientations can be orchestrated to arbitrary precision. This thesis develops the methods necessary for two such ensemble control problems arising in NMR, RF (control) amplitude inhomogeneity and systems with periodic drifts that exhibit dispersions in their amplitude and phase. In both cases, robust controls will rely on the non-commutativity of the system's dynamics enabling the generation of alternative and more robust control elements.