\mathbb{N}$ there exists $n> n_0$ such that $$\label{hypothesis-t}
\mathrm{Lie} \mathcal{V}_{n} = \mathfrak{su}(n).$$

\[metalemma-tracking\] Let $U_j=[-\delta,\delta]$ for some $\delta>0$ and every $j=1,\ldots,p$. Assume that $(\mathbb{A})$ holds true. If the [Lie–Galerkin Stalking Condition]{} holds then the system $$\dot{x} = (A+ u_1B_1+ \cdots + u_p B_p)x,\quad u\in U,$$ is a stalker.

If $U_{j} = [-\delta,\delta]$ for every $j=1,\ldots,p$, then the [Lie–Galerkin Stalking Condition]{} implies the [Lie–Galerkin Control Condition]{}, as it follows from the relation $$\left[ A^{(n)},\mathcal{E}_{\sigma}(B_{j}^{(n)}) \right] = \sigma J_{i}(\mathcal{E}_{\sigma}(B_{j}^{(n)})).$$

Controllability in higher norms {#sec:Hs}
-------------------------------

We define for $s > 0$, $$|A|^{s} \psi = \sum_{n\in {\mathbb{N}}} |\lambda_{n}|^{s} \langle \psi, \phi_{n}\rangle \phi_{n}$$ for every $\psi$ belonging to $$D(|A|^{s}) = \left\{ \psi \in {\mathcal{H}}\mid \sum_{n\in {\mathbb{N}}} |\lambda_{n}|^{2s} |\langle \psi, \phi_{n}\rangle|^{2} < +\infty  \right\}.$$ For every $\psi \in D(|A|^{s})$ we can define the $|A|^{s}$-norm (or simply $s$-norm) of $\psi$ by $\|\psi\|_{{s}} = \||A|^{s} \psi\|$. If $A$ is the Laplace–Dirichlet operator on some bounded domain of ${{\mathbb{R}}}^{n}$ then the $s$-norm is equivalent to the $H^{2s}$-norm on $D(|A|^{s})$.

Let $(A,B_1,\ldots,B_p, U, \Phi)$ satisfy Assumption $(\mathbb{A})$ and let $s>0$. System  is *approximately simultaneously controllable* (respectively *approximately controllable*) *for the $s$-norm* if for every $\varepsilon>0$, $r \in {\mathbb{N}}$ (respectively $r=1$), $\psi_1,\ldots,\psi_r$ in $D(|A|^{s})$, and $\hat{\Upsilon} \in \mathbf{U}({\mathcal{H}})$ such that 