Detectability of linear discrete-time systems with stochastic parameters

The paper introduces the concept of mean square detectability and relates this to the recently introduced concept of mean square observability. It is shown that under appropriate mean square detectability and stabilizability conditions the infinite-horizon optimal control problem for the general case of linear discrete time systems and quadratic criteria, both with stochastic parameters which are statistically independent of time, has a unique solution when the control system is mean square stable. A simple necessary and sufficient condition, explicit in the system matrices, is given to determine if a system is mean square detectable. This condition also holds for deterministic systems to be detectable in the usual sense. The mean square detectability property coincides with the usual one if the parameters are deterministic.