Optimal Design of Minimum Energy Pulses for Bloch Equations in the case of Dominant Transverse Relaxation

In this report, we apply Optimal Control Theory to design minimum energy $\pi \mathrm{/}2$ and $\pi$ pulses for Bloch equations, in the case where transverse relaxation rate is much larger than longitudinal so the later can be neglected. Using Pontryagin’s Maximum Principle, we derive an optimal feedback law and subsequently use it to obtain analytical expressions for the energy and duration of the optimal pulses.