Consider the scalar controllable linear system, whose state $X_{n}$ evolves by

$$X_{n+1}=X_{n}+U_{n}+\varepsilon_{n+1}$$

with observations $Y_{n}$ given by

$$Y_{n+1}=X_{n}+\eta_{n+1}$$

Here, $U_{n}$ is the control variable, which is to be determined on the basis of the observations up to time $n$, and $\varepsilon_{n}, \eta_{n}$ are independent $N(0,1)$ random variables. You wish to minimize the long-run average expected cost, where the instantaneous cost at time $n$ is $X_{n}^{2}+U_{n}^{2}$. You may assume that the optimal control in equilibrium has the form $U_{n}=-K \hat{X}_{n}$, where $\hat{X}_{n}$ is given by a recursion of the form

$$\hat{X}_{n+1}=\hat{X}_{n}+U_{n}+H\left(Y_{n+1}-\hat{X}_{n}\right)$$

and where $H$ is chosen so that $\Delta_{n}=X_{n}-\hat{X}_{n}$ is independent of the observations up to time $n$. Show that $K=H=(\sqrt{5}-1) / 2=2 /(\sqrt{5}+1)$, and determine the minimal long-run average expected cost. You are not expected to simplify the arithmetic form of your answer but should show clearly how you have obtained it.