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

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