## Abstract and Applied Analysis

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

#### 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 589562, 10 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412605759

Digital Object Identifier
doi:10.1155/2014/589562

Mathematical Reviews number (MathSciNet)
MR3193526

Zentralblatt MATH identifier
07022665

#### Citation

Yao, Fengqi; Cheng, Pei; Shen, Hao; Qiu, Li. On Input-to-State Stability of Impulsive Stochastic Systems with Time Delays. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 589562, 10 pages. doi:10.1155/2014/589562. https://projecteuclid.org/euclid.aaa/1412605759

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