are well-established topics in the quantum control of molecular dynamics both from the experimental and the theoretical point of view (see [@stapelfeldt; @sugny; @seideman] and references therein). For linear molecules driven by linearly polarized laser fields in gas phase, alignment means an increased probability direction along the polarization axis whereas orientation requires in addition the same (or opposite) direction as the polarization vector. A large amount of numerical simulations have been done in this domain but the mathematical part is not yet fully understood. From this perspective, the controllability problem is a necessary step towards comprehension.

We focus in this paper on the control by external fields of the rotation of a rigid linear molecule in ${{\mathbb{R}}}^3$. This control problem corresponds to the control of the Schrödinger equation on the unit sphere $S^2$. We show that the system driven by three fields along the three axes is approximately controllable for arbitrarily small controls.

Up to normalization of physical quantities (in particular, in units such that $\hbar=1$), the dynamics are governed by the equation $$\begin{aligned}
i\frac{\partial\psi(\theta,\varphi,t)}{\partial t}= -\Delta \psi(\theta,\varphi,t) + (u_1(t) \sin \theta\cos \varphi
+ u_2(t) \sin \theta\sin \varphi+ u_3(t) \cos \theta)\psi(\theta,\varphi,t)
\label{1}\end{aligned}$$where $\theta,\varphi$ are the spherical coordinates, which are related 