L2-exponential stability and impulsive stabilization of neutral stochastic delay differential equations

Abstract

This paper investigates $L^2$-exponential stability – also known as exponential stability in the mean square – of neutral stochastic delay differential equations with impulsive perturbations. Our primary objective is to stabilize an impulsive-free system by appropriately designing impulsive controls. Unlike previous studies, we introduce new and verifiable criteria for $L^2$-exponential stability. We further demonstrate that Euler–Maruyama-type approximations preserve $L^2$-exponential stability provided that the step sizes are sufficiently small; explicit conditions on these step sizes are derived. Moreover, we detail the design of impulsive perturbations that achieve $L^2$-exponential stabilization. Two examples are presented to validate the effectiveness of our criteria.