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Let $P=\mathbb{E}{\Delta }_n^2$ in equilibrium, with ${\Delta }_n=X_n-{\hat{X}}_n$. The innovation is $Z_{n+1}=Y_{n+1}-{\hat{X}}_n={\Delta }_n+{\eta }_{n+1}$. The given filter givesOrthogonality to the new innovation requires $(1-H)P-H=0$, hence $H=P/(P+1)$. Since the variables are jointly Gaussian, this orthogonality is independence. Stationary variance gives $P=(1-H{)}^{2}P+1+H^2=P/(P+1)+1$, soThe added process noise reflects the fact that the new observation concerns $X_n$, not $X_{n+1}$.

For the control calculation choose a quadratic value coefficient $R>0$. Conditional minimization of $x^2+u^2+R\mathbb{E}[(x+u+\epsilon {)}^2]$ gives $K=R/(1+R)$ and the Riccati equation $R=1+R-R^2/(1+R)$, hence $R^2=R+1$. Separation replaces $x$ by its conditional estimate, givingTo determine the average cost without solving an additional state covariance, complete the square:With $u=-K{\hat{X}}_n$, the final square is $K^2{\Delta }_n^2$. In equilibrium the expected telescoping value term is zero, so the minimal cost isThe identity also proves optimality: conditional mean minimization of the square uniquely selects the stated control. Here $1-K\in (0,1)$ ensures stable closed-loop estimates, making the equilibrium averaging valid. This is the Scalar delayed-observation LQG regulator.