| Abstract: |
| In this talk, we will present necessary and sufficient conditions for the controllability of Schr\odinger equations with electromagnetic potential. It is well known that such equations are controllable from an open set $\omega$ whenever it satisfies the geometric control condition; in that case, one has controllability for any time $T>0$. We show that this result can be improved on the sphere $\mathbb{S}^2$: in this setting, the system is controllable if $\omega$ satisfies a magnetic-field-dependent geometric control condition, which is weaker than the geometric control condition. Under this condition, there exists a minimum time, depending on the magnetic potential, beyond which the system is controllable. This geometric condition, and the minimum control time, are necessary for controllability as well. This is joint work with Gabriel Rivi\`ere (Nantes). |
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