Optimal design in the time domain is hard in general, but efficient approximation algorithms have been developed in some special cases. In this dissertation, we present new approaches to solving this problem using optimal control algorithms based on convex relaxations, and exploiting geometric structure in the underlying optimization problem.
Magnetic resonance imaging (MRI) serves as a motivating application problem throughout. We highlight two successes of these methods in the design of dynamic MRI experiments: magnetic resonance fingerprinting (MRF) for accelerated anatomic imaging, and hyperpolarized carbon-13 MRI for noninvasively monitoring cancer metabolism. In particular, we use optimal experiment design algorithms to compute optimized flip angle sequences for MRF and hyperpolarized carbon-13 acquisitions as well as optimized tracer injection inputs for estimating metabolic rate parameters in hyperpolarized carbon-13 acquisitions. In the final chapter, we present results on constrained reconstruction of metabolism maps from experimental data, closing the path from experiment design to data collection to synthesis of interpretable information.