  We consider the linear quadratic Gaussian control problem with a discounted
cost functional for descriptor systems on the infinite time horizon. Based on
recent results from the deterministic framework, we characterize the
feasibility of this problem using a linear matrix inequality. In particular,
conditions for existence and uniqueness of optimal controls are derived, which
are weaker compared to the standard approaches in the literature. We further
show that also for the stochastic problem, the optimal control is given in
terms of the stabilizing solution of the Lur'e equation, which generalizes the
algebraic Riccati equation.
