presented in this paper is more general than those present in the literature even in the single-input case. For instance it applies to the Laplace–Dirichlet operator on a compact interval of ${{\mathbb{R}}}$ with a control term of the type $u(t)x$. Let us mention that in [@ancoranoi], approximate simultaneous controllability of this model has been proved breaking the degeneracy between gaps among eigenstates through perturbation techniques. Here we prove the approximate simultaneous controllability and stalking without perturbation arguments. In this framework, proving a controllability result without perturbation arguments allows to translate directly the constructive proof of the main result in an algorithm that provides explicit expressions of controls [@Qtrack].

Brief description of the general results
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The main result of the paper is a sufficient condition for approximate simultaneous controllability which we call the [Lie–Galerkin Control Condition]{} (see Definition \[def:CCC\]).

Roughly speaking, both the sufficient condition proposed in [@ancoranoi] and the one presented here are based on the idea of driving the system with control laws in resonance with spectral gaps of the internal Hamiltonian $H_{0}$. However, whereas in [@ancoranoi] the only actions on the system obtained by resonance which are exploited for the controllability are those corresponding to elementary transitions 