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

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-infinity-loc[0,.infinity). 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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Contributor : François Chaplais <>
Submitted on : Thursday, April 22, 2010 - 6:50:25 PM
Last modification on : Thursday, April 15, 2021 - 3:31:39 AM
Long-term archiving on: : Tuesday, September 28, 2010 - 12:41:50 PM

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Karine Beauchard, Jean-Michel Coron, Pierre Rouchon. Controllability Issues for Continuous-Spectrum Systems and Ensemble Controllability of Bloch Equations. Communications in Mathematical Physics, Springer Verlag, 2010, 296 (2), pp.525-557. ⟨10.1007/s00220-010-1008-9⟩. ⟨hal-00475751⟩

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