Abstract and Applied Analysis
Volume 2013 (2013), Article ID 854743, 12 pages
http://dx.doi.org/10.1155/2013/854743
Research Article

Razumikhin-Type Theorems on Exponential Stability of SDDEs Containing Singularly Perturbed Random Processes

Junhao Hu,1 Xuerong Mao,2 and Chenggui Yuan3

1College of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China
2Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK
3Department of Mathematics, Swansea University, Swansea SA2 8PP, UK

Received 13 December 2012; Revised 25 April 2013; Accepted 25 April 2013

Academic Editor: Chuangxia Huang

Copyright © 2013 Junhao Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper concerns Razumikhin-type theorems on exponential stability of stochastic differential delay equations with Markovian switching, where the modulating Markov chain involves small parameters. The smaller the parameter is, the rapider switching the system will experience. In order to reduce the complexity, we will “replace” the original systems by limit systems with a simple structure. Under Razumikhin-type conditions, we establish theorems that if the limit systems are th-moment exponentially stable; then, the original systems are th-moment exponentially stable in an appropriate sense.