I have this example:
Let $n=2$ and $m=1$, $A=[-1,1]$, and write $x(t)=(x_1,x_2)^T$. Suppose
$ \left \{ \begin{matrix} x'_1=0 \\ x'_2=\alpha(t) \end{matrix} \right.$
This is the system of the form $x'=Mx+N\alpha$, for
$M= \left [ \begin{matrix} 0&0\\ 0&0 \end{matrix} \right]$, $N= \left [ \begin{matrix} 0\\ 1 \end{matrix} \right]$.
$C=$ set of initial points $x_0$ for which there exists a control such that $x(t)=0$ for some time $t$
The book says the next: Clearly $C=\{(x_1,x_2)|x_1=0\}$, the $x_2$-axis.
I don't understand because the set $C$ is this $\{(x_1,x_2)|x_1=0\}$.
Since $\dot{x}_1 = 0$, we see that $x_1(t) = x_1(0)$ for all $t$.
Hence if $x(t) = 0$ at some $t$, it is clear that we must have $x_1(0) = 0$, that is, $x(0) \in C$.
Furthermore, since the control $\alpha(t) = -\operatorname{sgn} x_2(0)$ will bring $x_2(t)$ to zero in some time, we see that if $x(0) \in C$ then there is a control such that $x(t) = 0$ for some $t$.