Maximov, I.I., Z. Tosner, and N.C. Nielsen, J. Chem. Phys., 128, (2008)
Optimal control theory has recently been introduced to nuclear magnetic resonance (NMR) spectroscopy as a means to systematically design and optimize pulse sequences for liquid- and solid-state applications. This has so far primarily involved numerical optimization using gradient-based methods, which allow for the optimization of a large number of pulse sequence parameters in a concerted way to maximize the efficiency of transfer between given spin states or shape the nuclear spin Hamiltonian to a particular form, both within a given period of time. Using such tools, a variety of new pulse sequences with improved performance have been developed, and the NMR spin engineers have been challenged to consider alternative routes for analytical experiment design to meet similar performance. In addition, it has lead to increasing demands to the numerical procedures used in the optimization process in terms of computational speed and fast convergence. With the latter aspect in mind, here we introduce an alternative approach to numerical experiment design based on the Krotov formulation of optimal control theory. For practical reasons, the overall radio frequency power delivered to the sample should be minimized to facilitate experimental implementation and avoid excessive sample heating. The presented algorithm makes explicit use of this requirement and iteratively solves the stationary conditions making sure that the maximum of the objective is reached. It is shown that this method is faster per iteration and takes different paths within a control space than gradient-based methods. In the present work, the Krotov approach is demonstrated by the optimization of NMR and dynamic nuclear polarization experiments for various spin systems and using different constraints with respect to radio frequency and microwave power consumption.