Root
/
Conference Proceedings
/
4th International Conference on Physics and Control (PhysCon 2009)
/
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 $-1/2$.

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 $-1/2$.