the converse is not necessarily true, since we restricted the set of admissible controls). Hence, the approximate simultaneous controllability of implies the approximate simultaneous controllability of . Moreover $$\|\tilde u_{j}\|_{L^{1}}  = \int_{0}^{t^{-1}(T)}| \tilde u_{j}(\tau)|d\tau =  
\int_{0}^{T}|u_{j}(t)|z(t) dt
= \int_{0}^{T}{|v_{j}(t)|}dt \leq  T,$$ for $j=1,\ldots,p$. The last inequality holds since either $U_{j} = [0,1]$ or $U_{j}=[-1,1]$. Hence the approximate simultaneous controllability in $\mathcal{L}$ of with a bound on the controllability time uniform with respect to the tolerance implies, in fact, the [$L^{1}$-bounded]{} approximate simultaneous controllability of .

Interaction framework {#sec:intfra}
---------------------

Given a solution $\psi(\cdot)$ of with controls $z(\cdot), v_{1}(\cdot), \ldots, v_{p}(\cdot)$ and a piecewise constant function $\alpha(\cdot)$ with values in $\{0,1\}$, let us define $$\omega(t) = \int_{0}^{t}(z(s)-\alpha(s))ds$$ and $$y(t) = e^{-\omega(t)A } \psi(t).$$ In particular $$\label{eq:1309}
|\langle \phi_k,y(t)\rangle | = |\langle \phi_k,\psi(t)\rangle|, \quad k\in {\mathbb{N}},$$ for every $t$. For $
\omega, v_{1},\ldots,v_{p} \in {{\mathbb{R}}}$ set $ \Theta(
\omega, v_{1},\ldots,v_{p})=
e^{-\omega A  }
(v_{1} {B_{1}} + \cdots + v_{p} {B_{p}} )
e^{\omega A }$. Note that $$\begin{aligned}
\Theta(\omega, v_{1},\ldots,v_{p})_{jk} &= \langle \phi_{k}, \Theta(\omega, v_{1},\ldots,v_{p}) \phi_{j} \rangle=
e^{i(\lambda_{k} - \lambda_{j})\omega} \left(v_{1}  (B_{1})_{jk}
+\cdots + v_{p} (B_{p})_{jk} \right),\label{m-Theta}\end{aligned}$$ and that $y(\cdot)$ satisfies $$\label{eq:main-interaction-pre}
\dot y(t)   = (\alpha(t) A+
\Theta(\omega(t), v_{1}(t),\ldots,v_{p}(t)))y(t), \qquad \alpha\in\{0,1\}, v\in U, 