$\hat\Upsilon \psi_{1}, \ldots, \hat\Upsilon \psi_{r} \in D(|A|^{s})$ there exists a piecewise constant function $u_\varepsilon: [0,T_\varepsilon]\to {{\mathbb{R}}}$ such that $$\|\hat{\Upsilon}\psi_j-\Upsilon_{T_\varepsilon}^{u_\varepsilon}
\psi_j\|_s<\varepsilon,$$ for every $j=1,\ldots,r$.

We say that $(A,B_{1}, \ldots, B_{p}, U, \Phi)$ satisfies $(\mathbb{A}')$ if it satisfies ($\mathbb{A}$) and the following additional assumptions hold:

-   the operator $i(A+u_{1}B_{1}+\cdots + u_{p}B_{p} )$ is bounded from below for every $u\in {{\mathbb{R}}}^{p}$;

-   the sequence $(\lambda_{k})_{k \in {\mathbb{N}}}$ is non-increasing and unbounded.

\[def:wc\] Let $(A,B_1,\ldots,B_p, U, \Phi)$ satisfy Assumption $(\mathbb{A}')$ and let $s>0$. Then $(A,B_1,\ldots,B_p)$ is *$s$-weakly-coupled* if $D(|A+u_{1}B_{1}+\cdots+ u_{p}B_{p}|^{s/2}) = D(|A|^{s/2})$ for every $u \in {{\mathbb{R}}}^{p}$ and there exists $C$ such that $$|\Re\langle |A|^{s} \psi,B_{l} \psi \rangle| \leq C|\langle |A|^{s}\psi, \psi\rangle|$$ for every $l=1,\ldots,p$, $\psi \in D(|A|^{s})$.

The following result is a consequence of [@weakly Proposition 2] and can be obtained by adapting the arguments proposed in [@weakly Proposition 5]. We provide its proof in Section \[6\].

\[thm:controlHs\] Let $(A,B_1,\ldots,B_p, U, \Phi)$ satisfy Assumption $(\mathbb{A}')$ and $(A,B_1,\ldots,B_p)$ be $s$-weakly coupled for some $s>0$. If  is [$L^{1}$-bounded]{} approximately simultaneously controllable then it is approximately simultaneously controllable for the $s/2$-norm.

As a direct consequence we have the following result generalizing [@weakly Proposition 5].

\[cor:qwerty\] Let $(A,B_1,\ldots,B_p, U, \Phi)$ satisfy Assumption $(\mathbb{A}')$ and $(A,B_1,\ldots,B_p)$ be $s$-weakly 