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Pré-Publication, Document De Travail Année : 2017

Regular propagators of bilinear quantum systems

Résumé

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]
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Dates et versions

hal-01016299 , version 1 (29-06-2014)
hal-01016299 , version 2 (26-10-2017)
hal-01016299 , version 3 (26-09-2019)

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Nabile Boussaid, Marco Caponigro, Thomas Chambrion. Regular propagators of bilinear quantum systems. 2017. ⟨hal-01016299v2⟩
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