Stability of delay differential equations with fading stochastic perturbations of the type of white noise and Poisson's jumps
This paper is mainly concerned with a delay differential equation with stochastic perturbations of the type of white noise and Poisson's jumps. They prove that if the stochastic perturbations decay on the infinity quickly enough then sufficient conditions for asymptotic stability of the zero solution of the deterministic differential equation with delay imply also asymptotic mean square stability of the zero solution of the stochastic differential equation. The stability conditions are obtained via the general method of Lyapunov functionals construction and the method of Linear Matrix Inequalities (LMIs). Investigation of the situation when stochastic perturbations do not fade on the infinity or fade not enough quickly is proposed as an unsolved problem.
scientific article; zbMATH DE number 7226746