SO(3) transformations are key elements in physics, such as for controlled spin systems. In experiments we have to consider a large ensemble of such physical entities, which are not exactly identical : we observe inhomogeneities of one or several physical parameters. Then, the spins do not interact identically with the control fields, and as a consequence, we can have various detrimental effects : small signal amplitude, low fidelity with the expected result…
Reducing the inhomogeneities is not always possible, but specific control fields can be designed to produce very selective transformations, or at the opposite very robust (insensitive) transformations for one or several physical parameters.
Optimal control theory can be used to design time-optimal selective and robust SO(3)-transformations. For this purpose, the Pontryagin Maximum Principle is applied to a model of two spins, which is simple enough for analytic computations and sufficiently complex to describe inhomogeneity effects. In this setting, selective and robust controls are respectively described by singular and regular trajectories. Using a geometric analysis combined with numerical simulations, the optimal solutions of different control problems are found. Selective and robust controls can be derived analytically without numerical optimization. The optimality of several standard control mechanisms in Nuclear Magnetic Resonance is shown, but new robust controls are also designed.
