Exponential Stability of Stochastic Systems with Delay and Poisson Jumps

Abstract

This paper focuses on the model of a class of nonlinear stochastic delay systems with Poisson jumps based on Lyapunov stability theory, stochastic analysis, and inequality technique. The existence and uniqueness of the adapted solution to such systems are proved by applying the fixed point theorem. By constructing a Lyapunov function and using Doob’s martingale inequality and Borel-Cantelli lemma, sufficient conditions are given to establish the exponential stability in the mean square of such systems, and we prove that the exponentially stable in the mean square of such systems implies the almost surely exponentially stable. The obtained results show that if stochastic systems is exponentially stable and the time delay is sufficiently small, then the corresponding stochastic delay systems with Poisson jumps will remain exponentially stable, and time delay upper limit is solved by using the obtained results when the system is exponentially stable, and they are more easily verified and applied in practice.

Existence and stability results for Caputo fractional stochastic differential equations with Lévy noise

“…Delay and Poisson jumps always coexist in real dynamic systems. Thus it is reasonable to consider them together, leading us to investigate the existence of solution of stochastic fractional delay differential equations with Lévy noise [36]. Let ξ(•) ∈ C[−δ, 0] be the initial path of x, where δ > 0 is a given finite time delay.…”

“…Exponential stability for stochastic neutral partial functional differential equations was obtained by Govindan using semigroup theory [13][14][15]. Zhu et al [36] studied the stability of stochastic systems with Poisson jumps. Further the stability of fractional dynamical systems is studied by many authors [10,16,30].…”