Regular propagators of bilinear quantum systems

The present analysis deals with the regularity of solutions of bilinear control systems of the type $x'=(A+u(t)B)x$where the state $x$ belongs to some complex infinite dimensional Hilbert space, the (possibly unbounded) linear operators $A$ and $B$ are skew-adjoint and the control $u$ is a real valued function. Such systems arise, for instance, in quantum control with the bilinear Schr\"{o}dinger equation. For the sake of the regularity analysis, we consider a more general framework where $A$ and $B$ are generators of contraction semi-groups.Under some hypotheses on the commutator of the operators $A$ and $B$, it is possible to extend the definition of solution for controls in the set of Radon measures to obtain precise a priori energy estimates on the solutions, leading to a natural extension of the celebrated noncontrollability result of Ball, Marsden, and Slemrod in 1982. Complementary material to this analysis can be found in [hal-01537743v1

Constructing packings in Grassmannian manifolds via alternating projection

This paper describes a numerical method for finding good packings in Grassmannian manifolds equipped with various metrics. This investigation also encompasses packing in projective spaces. In each case, producing a good packing is equivalent to constructing a matrix that has certain structural and spectral properties. By alternately enforcing the structural condition and then the spectral condition, it is often possible to reach a matrix that satisfies both. One may then extract a packing from this matrix. This approach is both powerful and versatile. In cases where experiments have been performed, the alternating projection method yields packings that compete with the best packings recorded. It also extends to problems that have not been studied numerically. For example, it can be used to produce packings of subspaces in real and complex Grassmannian spaces equipped with the Fubini--Study distance; these packings are valuable in wireless communications. One can prove that some of the novel configurations constructed by the algorithm have packing diameters that are nearly optimal.Comment: 41 pages, 7 tables, 4 figure

On Sussmann theorem for orbits of sets of vector fields on Banach manifolds

The purpose of this paper is to give some generalizations, in the context of Banach mani- folds, of Sussmann's results about the orbits of families of vector fields ([Su]). Essentially, we define the notion of "l1-orbits" for any family of vector fields on a Banach manifold, and we prove, under appropriate assumptions, that such an orbit is a weak Banach submanifold