Stability analysis of time-varying discrete stochastic systems with multiplicative noise and state delays

The aim of this paper is to study the stability of discrete stochastic time-delayed systems with multiplicative noise, where the coefficients are assumed to be time-varying with a general time-varying rate or a small time-varying rate. Firstly, by the Kronecker algebra theory and H-representation technique, the exponential stability of the stochastic system with common time-varying coefficients is investigated by the spectral approach. It is shown that the time-varying stochastic systems with state delays is exponentially stable in mean square sense if and only if its corresponding generalized spectral radius is less than one. Secondly, under definite conditions, by applying the so-called “frozen” technique, it is shown that the stability of a “frozen” system implies that of the corresponding slowly time-varying system.