Abstract:

We briefly overview some examples of quantum control problems
appearing in different fields, ranging from existing technology
(NMR/MRI) to research (control of nano devices, laser control of
molecules). In all these examples the same type of problem emerges: to
control a bilinear flow on a unitary (or orthogonal) group with
piecewise constant controls. We review the existing optimal control
approach, and point out some of its problems. Namely, the standard
extremal control equations can not generally hold for piecewise
constant controls; the only structural requirement from the controls
is to have small $L_2$ norm. We then discuss our approach which
derives the correct extremal equations, and which can prefer controls
with desired structure. Particularly, we are able to compute optimal
controls belonging to a desired subspace of control functions. Thus
our methods may hopefuly be useful for computing controls which are
producible by realistic laboratory lasers.

Time permitting, we shall also discuss an efficient discretization
scheme for Schrodinger equations based on Monge-Ampere PDEs. We
compute coordinate transformations, from a computational domain to the
physical one, which can zoom in or out of desired regions. In this way
computational effort is concentrated where it is needed.

(The work on control is joint with Antonella Zanna)

References
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1) I. Degani, A. Zanna, Lene Saelen, and Raymond Nepstad, Quantum
control with piecewise constant control functions, submitted (2008).

2) I. Degani, Monge-Ampere grids and the multidimensional mapped
Fourier method, journal of chemical physics, 128 (2008) 164108.