Certainty equivalence in stochastic linear time-lag systems

It is formally shown here that the optimal control for stochastic linear systems with quadratic criterion has the same form as the optimal control for the system obtained by replacing the random variables in the original system by their expected values conditioned on the measurements (certainty equivalence). It is also demonstrated that the separation theorem known for systems (with quadratic criterion) described by ordinary linear differential equations is valid if the system contains delayed state variables.