$p$-uple of piecewise constant controls $u(\cdot)=(u_{1}(\cdot), \ldots, u_{p}(\cdot))$ by composition of flows of the type $e^{t(A+u_{1}B_{1} + \cdots + u_{p} B_{p})}$. If, moreover, $B_{1},\ldots, B_{p}$ are bounded operators then the definition can be extended by continuity to every $L^{\infty}$ control law (see [@BMS Theorem 2.5]).

\[def:controllability\] Let $(A,B_{1}, \ldots, B_{p},U,\Phi)$ satisfy $(\mathbb{A})$. We say that is *approximately controllable* if for every $\psi_0,\psi_1$ in the unit sphere of $\cal H$ and every ${\varepsilon}>0$ there exists a piecewise constant control function $u:[0,T] \to U$ such that $
\|\psi_1-  {\Upsilon}^{u}_{T}(\psi_{0})\| <{\varepsilon}.
$

\[DEF\_simultaneous\_controllability\] Let $(A,B_{1}, \ldots, B_{p},U,\Phi)$ satisfy $(\mathbb{A})$. We say that is *approximately simultaneously controllable* if for every $r$ in $ {\mathbb{N}}$, $\psi_1,\ldots,\psi_{r}$ in ${\mathcal{H}}$, $\hat\Upsilon$ in $\mathbf{U}({\mathcal{H}})$, and ${\varepsilon}>0$ there exists a piecewise constant control $u:[0,T]\rightarrow U$ such that $$\left \| \hat\Upsilon \psi_k - {\Upsilon}^{u}_T \psi_k \right \|<{\varepsilon},\qquad k=1,\dots,r.$$ If, moreover, for every $\psi_1,\ldots,\psi_{r} \in \mathcal{L}$ and $\hat\Upsilon \in \mathbf{U}({\mathcal{H}})$ such that $\hat\Upsilon \psi_{1}, \ldots, \hat\Upsilon \psi_{r}  \in \mathcal{L}$, there exists $K>0$ (not depending on ${\varepsilon}$) such that $u$ can be chosen to satisfy, in addition, $\|u\|_{L^{1}} \leq K$, we say that is *[$L^{1}$-bounded]{} approximately simultaneously controllable*.

This last definition of controllability with *a priori* bound on the $L^{1}$-norm of the 