Spectral Perspective on the Stability of Discrete-Time Markov Jump Systems with Multiplicative Noise

We apply the spectrum analysis approach to address the stability of discrete-time Markov jump systems with state-multiplicative noise. In terms of the spectral distribution of a generalized Lyapunov operator, spectral criteria are presented to testify three different kinds of stochastic stabilities: asymptotic mean square stability, critical stability, and essential instability.