Exponential stability of impulsive stochastic functional differential equations

Abstract

In this paper, we investigate the pth moment and almost sure exponential stability of impulsive stochastic functional differential equations with finite delay by using Lyapunov method. Several stability theorems of impulsive stochastic functional differential equations with finite delay are derived. These new results are employed to impulsive stochastic equations with bounded time-varying delays and stochastically perturbed equations. Meanwhile, an example and simulations are given to show that impulses play an important role in pth moment and almost sure exponential stability of stochastic functional differential equations with finite delay.

Moment exponential stability of stochastic nonlinear delay systems with impulse effects at random times

“…The result in the aforementioned work states that under certain conditions and L k >0, the system is stable. However, the result displayed in the same work is different from ours because Theorem in our paper is a Razumikhin‐type theorem.…”

“…Remark The work of Pan and Cao considered the exponential stability for a class of stochastic functional differential equations. The result in the aforementioned work states that under certain conditions and L k >0, the system is stable. However, the result displayed in the same work is different from ours because Theorem in our paper is a Razumikhin‐type theorem.…”

“…In the work of Pan and Cao, the authors considered the p th moment exponential stability and almost sure exponential stability for a class of impulsive stochastic functional differential equations. It is worth noting that the findings in the aforementioned work regarding the p th moment exponential stability are different from the other literature referred to previously because they are not related to the Razumikhin‐type theorem. Recently, Zhu studied the well‐known Razumikhin‐type theorem for a class of stochastic functional differential equations with L vy noise and Markovian switching.…”

Exponential stability of stochastic functional differential equations with Markovian switching and delayed impulses via Razumikhin method

“…So the theory of impulsive differential equations is also attracting much attention in recent years [13][14][15][16][17][18][19]. Correspondingly, a lot of stability results of impulsive stochastic functional differential equations have been obtained [20][21][22][23][24][25][26]. However, there are few results on the stability of impulsive stochastic differential equation with Markovian switching.…”

“…The state variables on the impulses relate to the finite delay, which implies that the impulsive effects are more general than those given in [20,22,23]. Some Theorems on the pth moment exponential stability are derived in the case that the impulsive gain d ik +d ik < 1 or d ik +d ik ≥ 1.…”