I wish only to say that controllability and observabliity stand in a dual relationship to one another. This is clear in the necessary and sufficient conditions: that $[B\ AB\ \cdots A^{n-1}B]$ must be rank $n$ (for controllability), and
$\begin{bmatrix}C\\\ CA\\ \vdots\\ CA^{n-1}\end{bmatrix}$ must be of rank $n$ (for observability).
This duality is also nicely exhibited in the starred question on Example Sheet 3, Question 11. You should now be able to do this question.
Notice that a system can be stabilizable without being controllable. E.g. the scalar system with $x_{t+1}=(1/2)x_t$is trivially stabilizable, but it is not controllable.
$\begin{bmatrix}C\\\ CA\\ \vdots\\ CA^{n-1}\end{bmatrix}$ must be of rank $n$ (for observability).
This duality is also nicely exhibited in the starred question on Example Sheet 3, Question 11. You should now be able to do this question.
Notice that a system can be stabilizable without being controllable. E.g. the scalar system with $x_{t+1}=(1/2)x_t$is trivially stabilizable, but it is not controllable.
