Linear-quadratic-Gaussian control

Problem

Let $A$, $B$, $C$ be real matrices of sizes $n \times n$, $n \times m$ and $k \times n$, and let $T > 0$. Let $Q, F$ be symmetric positive semidefinite $n \times n$ matrices, and let $R$ be a symmetric positive definite $m \times m$ matrix. Let $W$ and $V$ be independent Wiener processes in $\R^n$ and $\R^k$ with covariance matrices $\Xi t$ and $\Theta t$. Here $\Xi$ is positive semidefinite and $\Theta$ is positive definite. Let the initial state $x(0)$ be Gaussian with mean $m_0$ and covariance $\Sigma_0$, independent of $W$ and $V$. The state $x$ and the observation $y$ solve the stochastic differential equations $$dx = (A x + B u) \, dt + dW, \qquad dy = C x \, dt + dV, \qquad y(0) = 0.$$ A control is admissible if $u(t)$ depends only on the observations $y(s)$, $s \le t$, that is, $u$ is adapted to the filtration generated by $y$. The linear-quadratic-Gaussian control problem is to minimize the expected value $$J(u) = \mathbb{E}\Big[ \int_0^T \big( x^{T} Q x + u^{T} R u \big) \, dt + x(T)^{T} F x(T) \Big]$$ over all admissible controls $u$.