My system has the linear form $x_t=Ax_{t-1}+Bu_{t-1}$. This is a model where the state $x_t$ is fully observable and there is no noise. $A$ is a $n\times n$ and $B$ is a $n\times m$ matrix. The cost function is $\sum_{t=0}^{h-1}Qu_t^2+x_h^2$.
Now I would like to argue, without simply putting everything in the Riccati equation that in terms of time to go $s$, $\Pi_s^{-1}$ obeys a linear recurrence and $$\Pi_s=[\frac{B^2}{Q(A^2-1)}+(1-\frac{B^2}{Q(A^2-1)})A^{-2s}]^{-1}$$
I do not see how to receive this term, may you can help me with that.