Exponential p-stability of impulsive stochastic differential equations with delays

Stability analysis of impulsive stochastic Cohen–Grossberg neural networks with mixed time delays

“…Consider the following model 21 (u 1 (t), u 1 (t − τ 21 (t)))dω 1 + σ 22 (u 2 (t), u 2 (t − τ 22 (t)))dω 2 , t = t k , ∆u 1 (t k ) = −(1 + 0.5 sin(1 + k))u 1 (t − ), ∆u 2 (t k ) = −(1 + 0.8 cos(2k 3 ))u 1 (t − ), (23) where t 0 = 0, t k = t k−1 + 0.5k, k = 1, 2, · · · , and g i (x) = f i (x) = h i (x) = x, τ ij (t) = 0.2| cos t| + 0.1, K ij (t) = te −t , i, j = 1, 2, is an M -matrix. On the anther hand, one can verify that (0, 0) T is an equilibrium point of model (23).…”

“…[21-25, 28-30, 33, 34] and the references therein. In particular, in [21,22], the authors have considered the exponential p-stability of impulsive stochastic differential equations with constant delays and obtained several stability conditions for checking the exponential p-stability. In [24,25,28,29,33,34], the stability of stochastic neural networks with constant or time-varying delay or bounded distributed delays have been considered and many interesting results have been established by employing a Lyapunov functional approach.…”

Controllability of nonlinear impulsive Ito type stochastic systems

“…Sufficient conditions for the stability of a class of nonlinear impulsive differential systems in a stochastic setting was obtained in (Rao and Tsokos, 1995). In (Yang et al, 2006), exponential stability of nonlinear impulsive stochastic differential equations with delays was established. More recently, Liu et al (2007) studied the existence, uniqueness and stability of stochastic impulsive systems using Lyapunov-like functions.…”

Asymptotic stability of neutral stochastic functional integro-differential equations with impulses

“…For example, non-autonomous and random dynamical systems perturbed by impulses are investigated in [20]. Jun Yang et al [22] studied the stability analysis of ISDEs with delays, Zhiguo Yang et al [23] analyzed the exponential p-th stability of ISDEs with delays, while in [17,18], R. Sakthivel and J. Luo investigated the existence and asymptotic stability in the p-th moment of mild solutions to ISDEs, with and without infinite delays, through the fixed point theory. Motivated by the works [14,15,17,18] we study in this paper the existence and asymptotic stability in the p-th moment of mild solutions of nonlinear neutral impulsive stochastic integro-differential equations (ISNIDEs) with delays under Lipschitz conditions.…”