On Lyapunov stability of scalar stochastic time-delayed systems

In this paper, we obtain an analytical Lyapunov- based stability conditions for scalar linear and nonlinear stochastic systems with discrete time-delay. The Lyapunov–Krasovskii and Lyapunov–Razumikhin methods are applied with techniques from stochastic calculus to obtain the regions of mean square asymptotic stability in the parameter space. Both delay-independent and delay-dependent stability conditions are analyzed corresponding to both additive and multiplicative stochastic Brownian motion excitation in the Ito form. It is also shown that the derived sufficient conditions are less conservative in comparison with other numerical LMI-based Lyapunov approaches. A range of different stability charts are obtained based on the derived Lyapunov-based stability criteria, which are also compared with numerical first and second moment stability boundaries computed using the stochastic semidiscretization method. A Lipschitz condition is used to treat nonlinear functions of the current and delayed states. © 2013, Springer-Verlag Berlin Heidelberg.