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Advances in Mathematical Physics
Volume 2012, Article ID 857493, 29 pages
Research Article

A Review of Geometric Optimal Control for Quantum Systems in Nuclear Magnetic Resonance

1Institut de Mathématiques de Bourgogne, UMR CNRS 5584, BP 47870, 21078 Dijon, France
2Department Chemie, Technische Universität München, Lichtenbergstrasse 4, 85747 Garching, Germany
3Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), UMR 5209 CNRS-Université de Bourgogne, 9 Avenue A. Savary, BP 47 870, 21078 Dijon Cedex, France

Received 30 June 2011; Revised 29 September 2011; Accepted 5 October 2011

Academic Editor: Ricardo Weder

Copyright © 2012 Bernard Bonnard et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We present a geometric framework to analyze optimal control problems of uncoupled spin 1/2 particles occurring in nuclear magnetic resonance. According to the Pontryagin's maximum principle, the optimal trajectories are solutions of a pseudo-Hamiltonian system. This computation is completed by sufficient optimality conditions based on the concept of conjugate points related to Lagrangian singularities. This approach is applied to analyze two relevant optimal control issues in NMR: the saturation control problem, that is, the problem of steering in minimum time a single spin 1/2 particle from the equilibrium point to the zero magnetization vector, and the contrast imaging problem. The analysis is completed by numerical computations and experimental results.