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## Section: Application Domains

### Quantum Control

The activity of B. Bonnard in quantum control started as a collaboration with D. Sugny (a physicist from ICB) in the ANR project Comoc, localized mainly at the University of Dijon; the problem was the control of the orientation of a molecule using a laser field, with a model that does take into account the dissipation due to the interaction with the environment, molecular collisions for instance. The model is a dissipative generalization of the finite dimensional Schrödinger equation, known as Lindblad equation. In particular we have computed the minimum time control and the minimum energy control for the orientation or a two-level system, using geometric optimal control and adapted numerical methods (shooting and numeric continuation)  [36] , [35] . The model is a 3-dimensional system depending upon 3 parameters, yielding a very complicated optimal control problem that we have solved for prescribed boundary conditions.

More recently, based on this project, we have reoriented our control activity towards Nuclear Magnetic Resonance (MNR). In MNR medical imaging, the contrast problem is the one of designing a variation of the magnetic field with respect to time that maximizes the difference, on the resulting image, between two different chemical species; this research is conducted with Prof. S. Glaser (TU-München); it was evidenced experimentally that the current contrast of the image is significantly improved by using “our” exact optimal control methods. The model is the Bloch equation for spin $\frac{1}{2}$ particles, that can be interpreted as a sub-case of Lindblad equation for a two-level system; the control problem to solve amounts to driving in minimum time the magnetization vector of the spin to zero (for parameters of the system corresponding to one of the species), and generalizations where such spin $\frac{1}{2}$ particles are coupled: double spin inversion for instance. This research project is supported on the french side by a PEPS INSIS (Control-Image) .