Time minimal saturation of a pair of spins and application in Magnetic Resonance Imaging
Bernard Bonnard, ,Institut de Mathématiques de Bourgogne, Université de Bourgogne, 9 avenue Alain Savary, 21078 Dijon, France, Olivier Cots, Jérémy Rouot, Thibaut Verron, ,INRIA, 2004 route des Lucioles F-06902, Sophia Antipolis, France, ,Toulouse Univ., INP-ENSEEIHT-IRIT, UMR CNRS 5505, 2 rue Camichel, 31071 Toulouse, France, ,EPF:École d'Ingénieur-e-s, 2 Rue F Sastre, 10430 Rosières-prés-Troyes, France · Mathematical Control and Related Fields · 2019
In this article, we analyze the time minimal control for the saturation of a pair of spins of the same species but with inhomogeneities of the applied RF-magnetic field, in relation with the contrast problem in Magnetic Resonance Imaging. 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. The ${\texttt {Bocop}}$ 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 computations combining shooting and continuation methods implemented in the ${\texttt {HamPath}}$ software to analyze the structure of the time minimal solution with respect to the set of parameters of the species. Symbolic computations techniques are used to handle the singularity analysis.