Let $x_0 \sim N(\mu_{x_0},\Sigma_{x_0})$, where $\mu_{x_0} \in \mathbb R^n$ and $\Sigma_{x_0} \in \mathbb R^{n \times n}$. Then, let $$ y_0 = Cx_0 + v_0 $$ where $C \in \mathbb R^{n \times n}, v_0 \sim N(0,\Sigma_{v_0})$, and $\Sigma_{v_0} \in \mathbb R^{n \times n}$. Suppose that $x_0$ and $v_0$ are independent, such that $y_0 \sim N(C\mu_{x_0},C^T \Sigma_{x_0}C + \Sigma_{v_0})$. Next, let $$ x_1 = Ax_0 + Bu_0(y_0) + w_0 $$ where
- $A \in \mathbb R^{n \times n}$,
- $B \in \mathbb R^{n \times n}$,
- $u_0 : \mathbb R^n \to \mathbb R^n$ is a vector-valued function of $y_0$, and
- $w_0 \sim N(0,\Sigma_{w_0})$ is independent of $x_0$ and $v_0$.
In Linear Quadratic Gaussian (LQG) control, we want to solve the following optimization problem $$ \min_{u_0} E_{x_0,w_0,v_0}[x_0^T Q_0 x_0 + u_0^T(y_0)R_0u_0(y_0) + x_1^TQ_1x_1] $$ where
- the notation $E_{x_0,w_0,v_0}[\cdot]$ indicates that we are averaging over all possible values of $x_0,w_0,$ and $v_0$, which are the fundamental sources of randomness in the problem,
- $Q_0$ and $Q_1$ are positive semi-definite matrices, and
- $R_0$ is a positive-definite matrix
In other words, we want to determine the optimal function $u_0$ that minimizes the above cost function. In the books that I've read that derive the solution to this optimization problem, they instead solve the following optimization problem $$ \min_{u_0} E_{x_0,x_1,y_0}[x_0^T Q_0 x_0 + u_0^T(y_0)R_0u_0(y_0) + x_1^TQ_1x_1] $$ Note that the expectation changed from $E_{x_0,w_0,v_0}[\cdot]$ to $E_{x_0,x_1,y_0}[\cdot]$. I'm not sure if these two optimization problems are equivalent. If they are, I'm not sure how to prove it.
