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Controllability Issues for Continuous-Spectrum Systems and Ensemble Controllability of Bloch Equations

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Abstract

We study the controllability of the Bloch equation, for an ensemble of non interacting half-spins, in a static magnetic field, with dispersion in the Larmor frequency. This system may be seen as a prototype for infinite dimensional bilinear systems with continuous spectrum, whose controllability is not well understood. We provide several mathematical answers, with discrimination between approximate and exact controllability, and between finite time or infinite time controllability: this system is not exactly controllable in finite time T with bounded controls in L 2(0, T), but it is approximately controllable in L ∞ in finite time with unbounded controls in \({L^{\infty}_{loc}([0,+\infty))}\). Moreover, we propose explicit controls realizing the asymptotic exact controllability to a uniform state of spin + 1/2 or −1/2.

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Authors and Affiliations

  1. CMLA, ENS Cachan, CNRS, Universud, 61 avenue du Président Wilson, F-94230, Cachan, France

    Karine Beauchard

  2. Institut Universitaire de France and Université Pierre et Marie Curie-Paris 6, UMR 7598 Laboratoire Jacques-Louis Lions, Paris, F-75005, France

    Jean-Michel Coron

  3. Mines ParisTech, Centre Automatique et Systèmes, Mathématiques et Systèmes, 60, boulevard Saint-Michel, 75272, Paris CEDEX, France

    Pierre Rouchon

Authors
  1. Karine Beauchard
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  2. Jean-Michel Coron
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  3. Pierre Rouchon
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Corresponding author

Correspondence to Karine Beauchard.

Additional information

Communicated by I.M. Sigal

KB, JMC and PR were partially supported by the “Agence Nationale de la Recherche” (ANR), Projet Blanc C-QUID number BLAN-3-139579.

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Beauchard, K., Coron, JM. & Rouchon, P. Controllability Issues for Continuous-Spectrum Systems and Ensemble Controllability of Bloch Equations. Commun. Math. Phys. 296, 525–557 (2010). https://doi.org/10.1007/s00220-010-1008-9

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  • DOI: https://doi.org/10.1007/s00220-010-1008-9

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