Consider an infinite-horizon classical Linear-Quadratic-Gaussian (LQG) problem \begin{eqnarray*} \underset{x,u}{\min} & & \underset{\eta\rightarrow\infty}{\lim}\mathbb{\mathbb{E}}\left[\frac{1}{\eta}\sum_{k=0}^{\eta-1}(x_{t}^{T}Qx_{t}+u_{t}^{T}Ru_{t})\right] \end{eqnarray*} for the discrete linear time invariant system \begin{eqnarray*} x_{t+1} & = & Ax_{t}+Bu_{t}+w_{t},\\ y_{t} & = & Cx_{t}+v_{t}, \end{eqnarray*} where $w_{t}\sim\mathcal{N}(0,W)$ and $v_{t}\sim\mathcal{N}(0,V)$ are independent sequences of iid zero-mean Gaussian random vectors with covariance matrices $W\succ0$ and $V\succ0$. The initial condition $x_{0}$ is a Gaussian random vector with mean $\overline{x}_{0}$ and covariance matrix $\overline{\Sigma}_{0},$ independent of the noise processes $w_{t}$ and $v_{t}$.
How can we transform the objective function \begin{eqnarray*} J & = & \underset{\eta\rightarrow\infty}{\lim}\mathbb{\mathbb{E}}\left[\frac{1}{\eta}\sum_{k=0}^{\eta-1}(x_{t}^{T}Qx_{t}+u_{t}^{T}Ru_{t})\right] \end{eqnarray*} as a function of the covariance matrix $\Sigma$ of of the estimation error please? I know that we can transform $J$ in something that looks like $Tr(Q\Sigma)+...$ Thanks