Consistency conditions for the asymptotic innovations representation and an equivalent inverse regulation problem

The consistency problem of time-invariant innovations representation is considered. Namely, given a time-invariant stochastic dynamic (conditionally) Gaussian system in the standard innovations representation form, when can it be truly considered an asymptotic innovation representation of observable input-output pairs? This problem is shown to be equivalent by duality to the following inverse regulation problem. Given a liner time-invariant system,when can a state-feedback input be considered a liner-quadratic optimal control law?