Communications in Mathematical Physics

, Volume 296, Issue 2, pp 525–557 | Cite as

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

  • Karine Beauchard
  • Jean-Michel Coron
  • Pierre Rouchon


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.


Functional Space Larmor Frequency Bloch Equation Exact Controllability Approximate Controllability 
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© Springer-Verlag 2010

Authors and Affiliations

  • Karine Beauchard
    • 1
  • Jean-Michel Coron
    • 2
  • Pierre Rouchon
    • 3
  1. 1.CMLA, ENS Cachan, CNRSUniversudCachanFrance
  2. 2.Institut Universitaire de France and Université Pierre et Marie Curie-Paris 6UMR 7598 Laboratoire Jacques-Louis LionsParisFrance
  3. 3.Mines ParisTech, Centre Automatique et SystèmesMathématiques et SystèmesParis CEDEXFrance

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