Geometric optimal control with an application to imaging in nuclear magnetic resonance

Abstract

This work addresses the contrast problem in nuclear magnetic resonance as a Mayer problem in optimal control. This is a problem motivated by improving the visible contrast in magnetic resonance imaging, in which the magnetization of the nuclei of the substances imaged are first prepared by being set to a particular configuration by an external magnetic field, the control. In particular we examine the contrast problem by saturation, wherein the magnetization of the first substance is set to zero. This system is modeled by a pair of Bloch equations representing the evolution of the magnetization vectors of the nuclei of two different substances, both influenced by the same control field. The Pontryagin maximum principle is used to reduce the problem to the analysis of so-called singular trajectories of the system, and we apply the tools of geometric optimal control. We explore the exceptional singular trajectories in detail. In this case the singular control, which is generically a feedback of the state and adjoint vectors of the Hamiltonian system, is a feedback of only the state for this problem, characterizing exceptional singular trajectories as solutions of an ordinary differential equation in the state variables. We introduce the concept of feedback equivalent control systems and results concerning quadratic differential equations, and compute a set of invariants for the quadratic approximation of exceptional singular ow to distinguish the different cases occurring in physical experiments. Additionally, Grobner bases are employed to make an algebraic-geometric classification of the equilibrium and singular points of the exceptional dynamics.