matrices $\mathcal{E}_\sigma(B_j^{(n)})$ for $(\sigma,j)\in\Xi_n$ correspond to “compatible dynamics” for the $n$-dimensional Galerkin approximation (*compatible*, that is, with higher dimensional Galerkin approximations).

Controllability results
-----------------------

Let $$\begin{aligned}
 \mathcal{V}_{n}^0 &= 
\left\{A^{(n)} 
\right\}
\cup \left\{{\cal E}_\sigma(B^{(n)}_j)\mid
(\sigma,j)\in \Xi_n 
\mbox{ and $j$ is such that }
(0,j)\in\Xi_n
\right\}\\
& \qquad 
\cup
\left\{ 
{\cal E}_\sigma(B^{(n)}_j)\mid
(\sigma,j)\in \Xi_n, \sigma \neq 0, U_{j} = [-\delta, \delta]
 \right\}.\end{aligned}$$

\[def:CCC\] Let $(A,B_{1}, \ldots, B_{p},U,\Phi)$ satisfy $(\mathbb{A})$. We say that the *[Lie–Galerkin Control Condition]{}* holds if for every $n_0\in \mathbb{N}$ there exists $n> n_0$ such that $$\label{hypothesis}
\mathrm{Lie} \mathcal{V}_{n}^0 \supseteq \mathfrak{su}(n).$$

\[metalemma\] Assume that $(\mathbb{A})$ holds true. If the [Lie–Galerkin Control Condition]{} holds then the system $$\dot{x} = (A+ u_1B_1+ \cdots + u_p B_p)x,\quad u\in U,$$ is [$L^{1}$-bounded]{} approximately simultaneously controllable.

Stalking results
----------------

For every $\xi \in S^{1} \subset {{\mathbb{C}}}$, consider the matrix operator $J_{\xi}$ such that $$\left(J_{\xi}(M)\right)_{j,k} = 
\begin{cases}
\xi M_{j,k} & \mbox{ if } \lambda_j<\lambda_k\\
0  & \mbox{ if }  \lambda_j=\lambda_k\\
\bar\xi M_{j,k} & \mbox{ if } \lambda_j>\lambda_k.
\end{cases}$$

Let $$ \mathcal{V}_{n} = 
\left\{J_{\xi}({\cal E}_\sigma(B^{(n)}_j))\mid
(\sigma,j)\in \Xi_n, \sigma \neq 0, \xi\in S^{1}
\right\}.$$

Notice that $\mathcal{V}_{n} \subset \mathfrak{su}(n)$.

\[def:CSC\] Let $(A,B_{1}, \ldots, B_{p},U,\Phi)$ satisfy $(\mathbb{A})$. We say that the *[Lie–Galerkin Stalking Condition]{}* holds if for every $n_0\in 