Exponential Stability of Impulsive Stochastic Functional Differential Equations With Delayed Impulses

ABSTRACT This paper focuses on the exponential stability of impulsive stochastic functional differential equations (ISFDEs). First, we successfully attempt to establish a criterion for the practical moment exponential stability (‐PES) of impulsive stochastic functional differential equations with delayed impulses (ISFDEDIs) using the newly established Halanay inequality. Furthermore, based on this inequality, we propose several sufficient conditions that have better applicability than the results in previous literature to ensure the system's moment exponential stability (‐ES). Next, we derive a novel result that guarantees the ‐ES of ISFDEDIs, based solely on the Lyapunov method. Additionally, using this approach, we establish a theorem on the generalized moment exponential stability (‐GES) for impulsive stochastic functional differential equations with unbounded delays (ISFDEUDs). Notably, these results allow the decay rate to be equal to or less than the upper bound of the gain, which can address the limitations of the aforementioned results and existing studies in certain special cases. Finally, we present several examples to validate the effectiveness of our findings.