$$x(t+1) =Ax(t)+Bu(t)=\left(\begin{matrix}0&0&0.8\\1&0&0\\0&1&0 \end{matrix}\right)x(t)+\left(\begin{matrix}1\\0\\0 \end{matrix}\right)u(t)$$ Basically the excercise is to find the minimum value of $N$ such that $$z^T\left(\displaystyle\sum_{\tau=0}^{N-1}A^\tau B B^T \left(A^T\right)^\tau \right)^{-1}z ≤ 1.1z^T\lim_{t \rightarrow \infty}\left(\displaystyle\sum_{\tau=0}^{t-1}A^\tau B B^T \left(A^T\right)^\tau \right)^{-1}z $$ for all $z$.
There is a hint: the matlab command P=dlyap(A,W) computes the solution of the Lyapunov equation $AP A^T + W = P,$ but I can't see how that helps.