Stability of Stochastic Differential Equations with State-dependent Delays

This chapter is devoted to stability investigation of systems with state-dependent delays under stochastic perturbations. Sufficient conditions of asymptotic mean square stability for the zero solution of a linear stochastic differential equation with distributed delays are obtained via the general method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). Besides delay-independent and delay-dependent conditions of stability in probability are obtained for two equilibria of a nonlinear stochastic differential equation with delay and exponential nonlinearity. The negative definiteness of matrices in the obtained LMIs is checked using the special MATLAB program. It is noted that the proposed research method can be used for the study of other types of linear and nonlinear systems with state-dependent delays. Numerical simulation of solutions of the considered stochastic differential equations with state-dependent delays illustrate the presented here theoretical results and open to readers attention a new unsolved problem of the obtained stability conditions improving.