Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance

Bernard Bonnard (1, 2) , Alain Jacquemard (2, 3) , Monique Chyba (4) , John Marriott (4, 1)

1 McTAO - Mathematics for Control, Transport and Applications
2 IMB - Institut de Mathématiques de Bourgogne [Dijon]
3 WPI - Wolfgang Pauli Institute
4 Department of Mathematics [Hawaii]

The analysis of the contrast problem in NMR medical imaging is essentially reduced to the analysis of the so-called singular trajectories of the system modeling the problem: a coupling of two spin 1/2 control systems. They are solutions of a constraint Hamiltonian vector field and restricting the dynamics to the zero level set of the Hamiltonian they define a vector field on B1 x B2, where B1 and B2 are the Bloch balls of the two spin particles. In this article we classify the behaviors of the solutions in relation with the relaxation parameters using the concept of feedback classification. The optimality status is analyzed using the feedback invariant concept of conjugate points.

Domaines: Optimisation et contrôle [math.OC]

Bernard Bonnard, Alain Jacquemard, Monique Chyba, John Marriott. Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance. Mathematical Control and Related Fields, 2013, 3 (4), pp.397-432. ⟨10.3934/mcrf.2013.3.397⟩. ⟨hal-00939495⟩