Stability of impulsive stochastic functional differential systems in terms of two measures via comparison approach

Abstract

Comparison principles for general impulsive stochastic functional differential systems are established. Employing the comparison principles and the theory of differential inequalities, stability and instability, involving two measures, of impulsive stochastic functional differential systems are investigated. Several stability and instability criteria are obtained, and two examples are also given to illustrate our results. Keywords stochastic systems, impulsive functional differential systems, comparison principle, stability in terms of two measures, instability Citation Yao F Q, Deng F Q. Stability of impulsive stochastic functional differential systems in terms of two measures via comparison approach.

Exponential Stability Analysis for Stochastic Delayed Differential Systems with Impulsive Effects: Average Impulsive Interval Approach

“…So conditions (i)-(iii) of Theorem 1 are satisfied with 1 = 2 = 1, c = −32.8081, c 1 = 3, d = −2 ln(1 + ) = 0.7133. Set = 2.14 and = 25.5, then > e c 1 > c. And consequently, applying 3 of Corollary 1, we conclude that the error dynamics (45) is mean square exponentially stable with impulse sequences satisfying(7) with N 0 ⩽ ∕d = 3, ⩽ d∕( − c) = 0.0122. Set = 0.01, N 0 = 3 and = 0.005, and construct the impulse sequence as { , 2 , N 0 , N 0 + , N 0 + 2 , 2N 0 , · · ·},…”

“…is pth moment exponential stable provided (7) holds with ⩽ d∕( −c) and N 0 ⩽ ∕d, where positive numbers and satisfy > e (3) is pth moment exponential stable for no impulse time sequence. Remark 2.…”

“…In the past decade, stochastic functional differential systems with impulsive effects (SFDSwIE), including stochastic delayed differential systems with impulsive effects (SDDSwIE), have received intensive attention and a great deal of stability results have been reported in the literature . However, nearly all these existing stability results are based on the infimum or supremum of impulsive intervals, that is, is bounded from below, or is bounded from above.…”

Stability in terms of two measures of solutions to stochastic partial differential delay equations with switching

“…Yuan investigated the stability in terms of two measures for stochastic differential equations with Markovian switching by using the method of Lyapunov functions. Yao and Deng in Yao and Deng and Yao and Deng discussed the stability in terms of two measures for impulsive stochastic functional differential systems via comparison approach. For the deterministic systems, there have been many researches about the stability in terms of two measures, Liu and Wang considered the stability in terms of two measures for impulsive systems of functional differential equations, Ahmad obtained sufficient conditions for the stability in terms of two measures for perturbed impulsive delay integro‐differential equations, Aleksandr and Slyn'ko discussed the stability in terms of two measures for a class of semilinear impulsive parabolic equations.…”

On Input-to-State Stability of Impulsive Stochastic Systems with Time Delays

“…Impulsive stochastic delayed systems incorporate impulses effects, stochastic perturbations, and time delays in one system simultaneously. During the last decade, there has been extensive interest in the study of force-free delayed impulsive stochastic systems; we refer to [24][25][26][27][28] and references therein. However, the corresponding theory for impulsive stochastic systems with external inputs has been relatively less developed.…”