Section: New Results
Optimal control for quantum systems and NMR
Participants : Bernard Bonnard, Mathieu Claeys [Imperial College, UK] , Olivier Cots, Thierry Combot, Pierre Martinon [project team COMMANDS] , Alain Jacquemard [Université de Bourgogne, IMB] .

The contrast imaging problem in nuclear magnetic resonance can be modeled as a Mayer problem, in the terminology of optimal control. The candidates as minimizers are selected among a set of extremals, solutions of a Hamiltonian system given by the Pontryagin Maximum Principle; sufficient second order conditions are known; they form the geometric foundations of the HAMPATH code which combines shooting and continuation methods.
In [4] , based on these theoretical studies, a thorough analysis of the case of deoxygenated/oxygenated blood samples is pursued, based on many numerical experiments.

We initiated more than a year ago a program to compare and study the complementarities between these methods based on the Pontryagin Maximum Principle are known as indirect methods,
 with the socalled direct methods where optimal control is seem as a generic optimization problem, as implemented in the Bocop software, developed in the COMMANDS projectteam,
 and with LMI techniques used to obtain global bounds on the extremum;
this was naturally done in collaboration with Pierre Martinon, an important contributor to Bocop and with Mathieu Claeys (LAAS CNRS, a PhD student supervised by J.B. Lasserre, now with Imperial College). The results are very promising, and there is a gain, numerically, in using both direct and indirect methods while working towards global optimality (in the contrast problem there are many local optima and the global optimality is a complicated issue). This is presented in [3] .
This also led to use algebraic techniques to further analyse the equations and their dependance of the materials to be discriminated [10] .

For time minimal control of a linear spin system with Ising coupling (more complex than the model above), we also analysed integrability properties of extremal solutions of the Pontryagin Maximum Principle, in relation with conjugate and cut loci computations. Restricting to the case of three spins, as in [11] , the problem is equivalent to analyze a family of almostRiemannian metrics on the sphere ${S}^{2}$ , with Grushin equatorial singularity. The problem can be lifted into a SRinvariant problem on $SO\left(3\right)$, this leads to a complete understanding of the geometry of the problem and to an explicit parametrization of the extremals using an appropriate chart as well as elliptic functions. This approach is compared with the direct analysis of the Liouville metrics on the sphere where the parametrization of the extremals is obtained by computing a Liouville normal form. This is backed by an algebraic approach applying differential Galois theory to integrability.