Stability and exact observability of discrete-time Markov jump systems with multiplicative noise

This study devotes to coping with stability and exact observability of discrete-time Markov jump systems subject to multiplicative noises. By employing a technique of spectrum, three important kinds of stabilities: asymptotic mean square stability, critical stability, and essential instability are first distinguished. Further, exact observability is introduced for the considered dynamical systems and a PBH criterion is presented in terms of the operator spectrum. Based on this criterion, the intrinsic relations among stability, exact observability and the solution of a generalized Lyapunov equation are fully addressed.