control achieving controllability has been observed in preceding works [@ancoranoi; @periodic]. It implies a stronger controllability property as shown in Section \[sec:Hs\].

Due to presence of the internal Hamiltonian and the boundedness of the controls, it is not possible in general to track, with arbitrarily precision, an unfeasible curve in $\mathcal{S}$. We introduce, then, the notion of *stalker*, that is a system for which it is possible to track any given curve up to phases (both for a single initial condition and in the spirit of simultaneous control). This definition makes sense from the physical point of view, since tracking up to phases means imposing the population of all energy levels of $H_{0}$ along the evolution.

The identification up to phases of elements of ${\mathcal{H}}$ in the basis $\Phi = (\phi_{k})_{k\in {\mathbb{N}}}$ can be introduced through the projection $${\mathfrak{M}}:\psi\mapsto \sum_{k\in {\mathbb{N}}}|\langle \phi_k,\psi\rangle|\phi_k.$$

\[def:trackable\] Let $(A,B_{1}, \ldots, B_{p},U,\Phi)$ satisfy $(\mathbb{A})$. We say that  is a *stalker* if for every $r$ in $ {\mathbb{N}}$, $\psi_1,\ldots,\psi_{r}$ in ${\mathcal{H}}$, $\hat\Upsilon:[0,T] \to \mathbf{U}({\mathcal{H}})$ continuous, with $\hat\Upsilon_{0} = \mathrm{Id}_{{\mathcal{H}}}$, and ${\varepsilon}>0$ there exist an invertible continuous function $\tau:[0,T] \to [0,T_{\tau}]$ and a piecewise constant control $u:[0,T_{\tau}]\rightarrow U$ such that $$\left \| {\mathfrak{M}}(\hat\Upsilon_{t} \psi_k) - {\mathfrak{M}}({\Upsilon}^{u}_{\tau(t)} \psi_k) 