The system is not completely reachable, Is state x1 reachable from x0?
Since the rank is 2, we will need a minimum of 2 steps.

Let's take them to be the first two \$u_0\$ and \$u_1\$. This gives
$$x_1=B u_0$$
$$\ \ \ \ \ \ \ \ \ \ \ \ \ x_2=B u_1+A B u_0$$
Substituting values
$$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(
\begin{array}{c}
 3 \\
 2 \\
 2 \\
\end{array}
\right) = \left(
\begin{array}{cc}
 0 &amp; 1 \\
 1 &amp; 1 \\
 1 &amp; 1 \\
\end{array}
\right)\left(
\begin{array}{c}
 u_1 \\
 u_0 \\
\end{array}
\right)$$

The solution is \$u_0=3, u_1=-1\$. I got this by observation, but in general you can solve the above using the pseudo inverse.

(For lower order systems, you can also visualize these controllable subspaces. I had created some earlier for linear and nonlinear continuous time systems. Another interesting one.)