Geometric and numerical methods for the contrast and saturation problems in Magnetic Resonance Imaging

In this talk, we present the time minimal control problem about the saturation of a pair of spins of the same species but with inhomogeneities on the applied RF-magnetic field, in relation with the contrast problem in MRI. We make a complete analysis based on geometric control to classify the optimal syntheses in the single spin case, to pave the road to analyze the case of two spins. This points out the phenomenon of bridge, which consists in linking two singular arcs by a bang arc to bypass some singularities of the singular extremal flow. In the case of two spins, the question about global optimality is more intricate. The Bocop software is used to determine local minimizers for physical test cases and Linear Matrix Inequalities approach is applied to estimate the global optimal value and validate the previous computations. This is complemented by numerical investigations combining shooting and continuation methods implemented in the HamPath software to analyze the structure of the time minimal solution with respect to the set of parameters of the species. This reveals new complex structures, compare to the single spin case.