Periodic Averaging Principle for Neutral Stochastic Delay Differential Equations with Impulses

Peiguang Wang,Yan Xu

doi:10.1155/2020/6731091

Peiguang Wang, Yan Xu

Open Access

https://doi.org/10.1155/2020/6731091

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Journal: Complexity	Publication Date: Jun 15, 2020
Citations: 2	License type: CC BY 4.0

Affiliation: Hebei University

Abstract

In this paper, we study the periodic averaging principle for neutral stochastic delay differential equations with impulses under non-Lipschitz condition. By using the linear operator theory, we deal with the difficulty brought by delay term of the neutral system and obtain the conclusion that the solutions of neutral stochastic delay differential equations with impulses converge to the solutions of the corresponding averaged stochastic delay differential equations without impulses in the sense of mean square and in probability. At last, an example is presented to show the validity of the proposed theories.

Highlights

Delay, impulse, and noise are natural phenomena in most practical issues and those phenomena are generally modeled by stochastic delay differential equations with impulses, and noise can be described by Brownian motion
It allows to avoid the detailed analysis of complex original systems and consider the simplified equations. e first analysis for averaging principle for stochastic differential equations was deeply addressed by Khasminskij [1]
We overcome the difficulties caused by the delay term which is included under the differentiation at the left-hand side, and the impulse appears in neutral stochastic differential equations

Summary

Introduction

Impulse, and noise are natural phenomena in most practical issues and those phenomena are generally modeled by stochastic delay differential equations with impulses, and noise can be described by Brownian motion. Because of the complexity of the system, it is difficult to obtain the exact solution of the vast majority of stochastic delay differential equations with impulses In this background, it is very important to look for an approximate system which is more amenable for analysis and simulation, and it governs the evolution of the original system over a long time scale. E averaging principle is an effective method to understand the main part of the behavior of dynamical systems It allows to avoid the detailed analysis of complex original systems and consider the simplified equations. Averaging principle for stochastic delay differential equations of the neutral type driven by G-Brownian motion was studied [10], in which the impulses are not considered in the system.

Model Description and Preliminaries

Main Results and Proofs

Illustrative Example

Conclusion