Exact detectability and exact observability of discrete-time linear stochastic systems with periodic coefficients

Abstract

This paper is devoted to the problem of detectability and observability of a class of time-varying stochastic systems. The main contribution of this note is to propose a PBH-type test which is equivalent to the detectability of the considered stochastic systems in the sense that all unstable modes produce some non-zero output. Then, we apply the obtained results in order to derive a criterion of exponential stability in mean square sense as well as existence conditions for the stabilizing solution of a class of discrete-time time-varying Riccati equations.