Simple LMIs for stability of stochastic systems with delay term given by Stieltjes integral or with stabilizing delay

This paper introduces stability conditions in the form of linear matrix inequalities (LMIs) for general linear retarded systems with a delay term described by Stieltjes integral. The derived LMIs provide in the unified form conditions for both discrete and distributed delays. Two Lyapunov-based methods for the asymptotic mean square stability of stochastic linear time-invariant systems are presented. The first one employs neutral type model transformation and augmented Lyapunov functionals. Differently from the existing LMI stability conditions based on neutral type transformation, the proposed conditions do not require the stability of the corresponding integral equations. Moreover, it is shown that in the simplest existing LMIs based on non-augmented Lyapunov functionals, the stability analysis of the integral equation can be omitted. The second method is based on a stochastic extension of simple Lyapunov functionals depending on the state derivative. The same two methods are further applied to delay-induced stability analysis of stochastic vector second-order systems, simplifying the recent results via neutral type transformation and leading to new conditions for stochastic systems via the second method. Numerical examples give comparison of results via different methods.