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Controllability issues for continuous-spectrum systems and ensemble controllability of Bloch Equations

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^\infty 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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