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Abstract and Applied Analysis
Volume 2014, Article ID 589562, 10 pages
http://dx.doi.org/10.1155/2014/589562
Research Article

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

1School of Electrical Engineering & Information, Anhui University of Technology, Maanshan 243000, China
2School of Mathematical Science, Anhui University, Hefei 230039, China
3College of Mechatronics and Control Engineering, Shenzhen University, Shenzhen 518060, China

Received 6 January 2014; Accepted 1 March 2014; Published 3 April 2014

Academic Editor: Shuping He

Copyright © 2014 Fengqi Yao 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 is concerned with pth moment input-to-state stability (p-ISS) and stochastic input-to-state stability (SISS) of impulsive stochastic systems with time delays. Razumikhin-type theorems ensuring p-ISS/SISS are established for the mentioned systems with external input affecting both the continuous and the discrete dynamics. It is shown that when the impulse-free delayed stochastic dynamics are p-ISS/SISS but the impulses are destabilizing, the p-ISS/SISS property of the impulsive stochastic systems can be preserved if the length of the impulsive interval is large enough. In particular, if the impulse-free delayed stochastic dynamics are p-ISS/SISS and the discrete dynamics are marginally stable for the zero input, the impulsive stochastic system is p-ISS/SISS regardless of how often or how seldom the impulses occur. To overcome the difficulties caused by the coexistence of time delays, impulses, and stochastic effects, Razumikhin techniques and piecewise continuous Lyapunov functions as well as stochastic analysis techniques are involved together. An example is provided to illustrate the effectiveness and advantages of our results.