Abstract and Applied Analysis

Exponential Stability of Impulsive Stochastic Functional Differential Systems

Zheng Wu, Hao Huang, and Lianglong Wang

Full-text: Open access

Abstract

This paper is concerned with stabilization of impulsive stochastic delay differential systems. Based on the Razumikhin techniques and Lyapunov functions, several criteria on pth moment and almost sure exponential stability are established. Our results show that stochastic functional differential systems may be exponentially stabilized by impulses.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 678536, 12 pages.

Dates
First available in Project Euclid: 1 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364846454

Digital Object Identifier
doi:10.1155/2012/678536

Mathematical Reviews number (MathSciNet)
MR2922930

Zentralblatt MATH identifier
1242.93109

Citation

Wu, Zheng; Huang, Hao; Wang, Lianglong. Exponential Stability of Impulsive Stochastic Functional Differential Systems. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 678536, 12 pages. doi:10.1155/2012/678536. https://projecteuclid.org/euclid.aaa/1364846454


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References

  • J. Baštinec, J. Diblík, D. Y. Khusainov, and A. Ryvolová, “Exponential stability and estimation of solutions of linear differential systems of neutral type with constant coefficients,” Boundary Value Problems, vol. 2010, Article ID 956121, 20 pages, 2010.
  • A.V. Shatyrko, D. Y. Khusainov, J. Diblík, J. Bastinec, and A. Ryvolova, “Estimates of perturbations of nonlinear indirect interval control system of neutral type,” Journal of Automation and Information Sciences, vol. 43, no. 1, pp. 13–28, 2011.
  • S. Peng and L. Yang, “Global exponential stability of impulsive functional differential equations via Razumikhin technique,” Abstract and Applied Analysis, vol. 2010, Article ID 987372, 11 pages, 2010.
  • J. Diblík and A. Zafer, “On stability of linear delay differential equations under Perron's condition,” Abstract and Applied Analysis, vol. 2011, Article ID 134072, 9 pages, 2011.
  • J. Diblík, D. Y. Khusainov, I. V. Grytsay, and Z. Šmarda, “Stability of nonlinear autonomous quadratic discrete systems in the critical case,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 539087, 23 pages, 2010.
  • J. Diblík, D. Ya. Khusainov, and I.V. Grytsay, “Stability investigation of nonlinear quadratic discrete dynamics systems in the critical case,” Journal of Physics: Conference Series, vol. 96, no. 1, Article ID 012042, 2008.
  • J. Luo, “Exponential stability for stochastic neutral partial functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 355, no. 1, pp. 414–425, 2009.
  • I. A. Dzhalladova, J. Baštinec, J. Diblík, and D. Y. Khusainov, “Estimates of exponential stability for solutions of stochastic control systems with delay,” Abstract and Applied Analysis, vol. 2011, Article ID 920412, 14 pages, 2011.
  • S. Janković, J. Randjelović, and M. Jovanović, “Razumikhin-type exponential stability criteria of neutral stochastic functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 355, no. 2, pp. 811–820, 2009.
  • Z. Yu, “Almost surely asymptotic stability of exact and numerical solutions for neutral stochastic pantograph equations,” Abstract and Applied Analysis, vol. 2011, Article ID 143079, 14 pages, 2011.
  • X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, UK, 1997.
  • V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6, World Scientific Publishing, Teaneck, NJ, USA, 1989.
  • A. Lin and L. Hu, “Existence results for impulsive neutral stochastic functional integro-differential inclusions with nonlocal initial conditions,” Computers & Mathematics with Applications, vol. 59, no. 1, pp. 64–73, 2010.
  • B. Liu, “Stability of solutions for stochastic impulsive systems via comparison approach,” IEEE Transactions on Automatic Control, vol. 53, no. 9, pp. 2128–2133, 2008.
  • R. Sakthivel and J. Luo, “Asymptotic stability of nonlinear impulsive stochastic differential equations,” Statistics & Probability Letters, vol. 79, no. 9, pp. 1219–1223, 2009.
  • Q. Song and Z. Wang, “Stability analysis of impulsive stochastic Cohen-Grossberg neural networks with mixed time delays,” Physica A, vol. 387, no. 13, pp. 3314–3326, 2008.
  • X. Wang, Q. Guo, and D. Xu, “Exponential p-stability of impulsive stochastic Cohen-Grossberg neural networks with mixed delays,” Mathematics and Computers in Simulation, vol. 79, no. 5, pp. 1698–1710, 2009.
  • H. Wu and J. Sun, “p-moment stability of stochastic differential equations with impulsive jump and Markovian switching,” Automatica, vol. 42, no. 10, pp. 1753–1759, 2006.
  • L. Xu and D. Xu, “Mean square exponential stability of impulsive control stochastic system with time-varying delay,” Physics Letters A, vol. 373, no. 3, pp. 328–333, 2009.
  • L. Shen and J. Sun, “p-th moment exponential stability of stochastic differential equations with im-pulse effect,” Science China Information Sciences, vol. 54, no. 8, pp. 1702–1711, 2011.
  • P. Cheng, F. Deng, and X. Dai, “Razumikhin-type theorems for asymptotic stability of impulsive stochastic functional differential systems,” Journal of Systems Science and Systems Engineering, vol. 19, no. 1, pp. 72–84, 2010.
  • S. Peng and B. Jia, “Some criteria on pth moment stability of impulsive stochastic functional dif-ferential equations,” Statistics & Probability Letters, vol. 80, no. 13-14, pp. 1085–1092, 2010.
  • P. Cheng and F. Deng, “Global exponential stability of impulsive stochastic functional differential systems,” Statistics & Probability Letters, vol. 80, no. 23-24, pp. 1854–1862, 2010.
  • J. Liu, X. Liu, and W.-C. Xie, “Impulsive stabilization of stochastic functional differential equations,” Applied Mathematics Letters, vol. 24, no. 3, pp. 264–269, 2011.
  • J. Liu, X. Liu, and W.-C. Xie, “Existence and uniqueness results for impulsive hybrid stochastic delay systems,” Communications on Applied Nonlinear Analysis, vol. 17, no. 3, pp. 37–53, 2010.