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I have the following $(A,B,C)$ state-space sytem:

$$ A = \begin{bmatrix} -2 & 0 & 0 \\ -1 & -1 & 2 \\ -1 & 0 & 0 \\ \end{bmatrix},\ B = \begin{bmatrix} 0 \\ 1 \\ 0 \\ \end{bmatrix},\ C = \begin{bmatrix} 1 & 0 & 1 \end{bmatrix} $$

My question is how do I compute if a point, e.g. $\begin{bmatrix} 0 & 2 & 1\end{bmatrix}^T$, in state space can be reached from $0$ by the choice of an input signal $u(t)$?

Note: the system is not controllable and not observable.

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Compute the controllability

$$ C = \begin{bmatrix} B & AB & A^2B \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \\ 1 & -1 & 1 \\ 0 & 0 & 0 \end{bmatrix} $$

Hence only $x_2$ can be controlled thus only states $\begin{bmatrix} 0 \\ \alpha \\ 0 \end{bmatrix}$ can be reached.

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