Stability of impulsive stochastic differential equations with Markovian switching

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

“…To date, a large number of works on impulsive stochastic differential equations and impulsive stochastic functional differential equations have appeared in the literature. For example, Xu and He used the matrix inequality technique and derived some sufficient conditions to ensure the exponential stability for a class of impulsive stochastic differential equations. Moreover, Ren et al investigated the stability for impulsive stochastic differential equations driven by the G‐Brownian motion.…”

Exponential stability of hybrid stochastic functional differential systems with delayed impulsive effects: average impulsive interval approach

“…It is worth noting that by using Lyapunov function and Razumikhin techniques, some sufficient conditions for the pth moment exponential stability of system (23) have been established in [21][22][23][24][25], while the existing stability results in [21][22][23][24][25] are based on the infimum or supremum of impulsive intervals and an appropriate minimal class of functionals relative to which the derivative of Lyapunov function or Lyapunov functional is estimated, which could be rather conservative and not convenient. Thus, our results in Corollary 3.1 are better than theirs to some extent.…”

“…Assume that there exist symmetrical positive matrixes P 1 , P 2 , : : :, P N and several constants d 1 > 0, d 2 > 0, % > 0, Q Á > 0, N 0 2 N, and Á 2 R such that (i) for all i, j, l, m, q, s 2 S, the following matrix inequalities hold: 2 Then system (25) is exponentially stable in the mean square, and the convergence rate should not be greater than 2 , where is the unique positive solution of C Á C ln.d1C˛1d2/ % CˇQ Áe D 0.…”

“…On the other hand, impulsive effects may exist widely in many evolutionary processes in which they experienced changes of state abruptly at certain moments in the fields such as economics, medicine, and biology (see, e.g., [11][12][13][14][15][16][17][18]). More recently, some results on impulsive stochastic functional equations with Markovian switching have been reported in [19][20][21][22][23][24][25] . For example, robust exponential stability of impulsive stochastic functional equations with Markovian switching has been investigated in [19,20].…”

On hybrid stochastic population models with impulsive perturbations

“…As far as we know, there was no much contribution to this kind of stochastic differential equation. The authors in [30,31] studied the impulsive perturbations at the switching points of a hybrid stochastic differential equation with impulsive perturbations. This motivated us to investigate some properties of (4).…”