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

Author

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  • Fengqi Yao
  • Pei Cheng
  • Hao Shen
  • Li Qiu

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.

Suggested Citation

  • Fengqi Yao & Pei Cheng & Hao Shen & Li Qiu, 2014. "On Input‐to‐State Stability of Impulsive Stochastic Systems with Time Delays," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:589562
    DOI: 10.1155/2014/589562
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    References listed on IDEAS

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    1. Fengqi Yao & Feiqi Deng & Pei Cheng, 2013. "Exponential Stability of Impulsive Stochastic Functional Differential Systems with Delayed Impulses," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-8, April.
    2. Yao, Fengqi & Deng, Feiqi, 2012. "Exponential stability in terms of two measures of impulsive stochastic functional differential systems via comparison principle," Statistics & Probability Letters, Elsevier, vol. 82(6), pages 1151-1159.
    3. Fengqi Yao & Feiqi Deng & Pei Cheng, 2013. "Exponential Stability of Impulsive Stochastic Functional Differential Systems with Delayed Impulses," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
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