$u_3$. Secondly, the general theory developed in [@noi; @nersesyan; @ancoranoi] is based on nonresonance conditions on the spectrum of the internal Hamiltonian. The Laplace–Beltrami operator on $S^2$, however, has a severely degenerate spectrum, since the $\ell$-th eigenvalue $-i\ell(\ell+1)$ has multiplicity $2\ell+1$. In [@noi] we proposed a perturbation technique in order to overcome resonance relations in the spectrum of the drift. This technique was applied in [@noicdc] to the case of the orientation of a molecule confined in a plane driven by one control. The planar case is already technically challenging and a generalization of the same technique to the case of three controls in 3D seems hard to achieve.

Structure of the paper
----------------------

The structure of the paper is the following: in the next section we present the general multi-input abstract framework and the main abstract results. In Section \[rotational\] we apply them to system . The proofs of the abstract results are contained in Sections \[sec:prova1\], \[sec:tracking\], and \[6\].

Framework and main results {#sec:mainresults}
==========================

Let $p\in {\mathbb{N}}$, $\delta>0$, and $U=U_1\times \cdots \times U_p$ with either $U_j=[0,\delta]$ or $U_j=[-\delta,\delta]$.

\[def:system\] Let $\cal H$ be an infinite-dimensional Hilbert space with scalar product $\langle \cdot ,\cdot \rangle$ and $A,B_{1}, \ldots,B_{p}$ 