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arXiv:1303.0298

A Sufficient Condition for Partial Ensemble Controllability of Bilinear Schrödinger Equations with Bounded Coupling Terms

Thomas Chambrion

Published Mar 1, 2013• 0 citations
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Abstract

This note presents a sufficient condition for partial approximate ensemble controllability of a set of bilinear conservative quantum systems in an infinite dimensional Hilbert space. The proof relies on classical geometric and averaging control techniques applied on finite dimensional approximation of the infinite dimensional system. The results are illustrated with the planar rotation of a linear molecule.

DOI
10.48550/arxiv.1303.0298

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This note presents a sufficient condition for partial approximate ensemble controllability of a set of bilinear conservative quantum systems in an infinite dimensional Hilbert space. The proof relies on classical geometric and averaging control techniques applied on finite dimensional approximation of the infinite dimensional system. The results are illustrated with the planar rotation of a linear molecule.
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In the framework of bilinear control of the Schrödinger equation with bounded control operators, it has been proved that the reachable set has a dense complemement in S∩H2{\cal S}\cap {\cal H}^{2}S∩H2. Hence, in this setting, exact quantum control in infinite dimensions is not possible. On the other hand it is known that there is a simple choice of operators which, when applied to an arbitrary state, generate dense orbits in Hilbert space. Compatibility of these two results is established in this paper and, in particular, it is proved that the closure of the reachable set of bilinear control by bounded operators is dense in S∩H2{\cal S}\cap {\cal H}^{2}S∩H2. The requirements for controllability in infinite dimensions are also related to the properties of the infinite dimensional unitary group.