Mathematics – Optimization and Control
Scientific paper
2009-03-16
Commun. Math. Phys., Volume 296, Number 2, June 2010, p.525-557
Mathematics
Optimization and Control
submitted
Scientific paper
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.
Beauchard Karine
Coron Jean-Michel
Rouchon Pierre
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