## Abstract and Applied Analysis

### Stability Analysis of Impulsive Stochastic Functional Differential Equations with Delayed Impulses via Comparison Principle and Impulsive Delay Differential Inequality

#### Abstract

The problem of stability for nonlinear impulsive stochastic functional differential equations with delayed impulses is addressed in this paper. Based on the comparison principle and an impulsive delay differential inequality, some exponential stability and asymptotical stability criteria are derived, which show that the system will be stable if the impulses’ frequency and amplitude are suitably related to the increase or decrease of the continuous stochastic flows. The obtained results complement ones from some recent works. Two examples are discussed to illustrate the effectiveness and advantages of our results.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 710150, 9 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412278119

Digital Object Identifier
doi:10.1155/2014/710150

Mathematical Reviews number (MathSciNet)
MR3173285

Zentralblatt MATH identifier
07022924

#### Citation

Cheng, Pei; Yao, Fengqi; Hua, Mingang. Stability Analysis of Impulsive Stochastic Functional Differential Equations with Delayed Impulses via Comparison Principle and Impulsive Delay Differential Inequality. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 710150, 9 pages. doi:10.1155/2014/710150. https://projecteuclid.org/euclid.aaa/1412278119

#### References

• V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Singapore, 1989.
• A. M. Samoilenko and M. O. Perestyk, Impulsive Differential Equations, vol. 28 of Series on Advances in Mathematics for Applied Sciences, World Scientific, Singapore, 1995.
• K. Gopalsamy and B. G. Zhang, “On delay differential equations with impulses,” Journal of Mathematical Analysis and Applications, vol. 139, no. 1, pp. 110–122, 1989.
• Y. V. Rogovchenko, “Impulsive evolution systems: main results and new trends,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 3, no. 1, pp. 57–88, 1997.
• G. Ballinger and X. Z. Liu, “Existence and uniqueness results for impulsive delay differential equations,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 5, no. 1–4, pp. 579–591, 1999.
• G. Ballinger and X. Z. Liu, “Existence, uniqueness and boundedness results for impulsive delay differential equations,” Applicable Analysis, vol. 74, no. 1-2, pp. 71–93, 2000.
• W. M. Haddad, V. S. Chellaboina, and S. G. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, USA, 2006.
• A. V. Anokhin, L. Berezansky, and E. Braverman, “Exponential stability of linear delay impulsive differential equations,” Journal of Mathematical Analysis and Applications, vol. 193, no. 3, pp. 923–941, 1995.
• I. M. Stamova and G. T. Stamov, “Lyapunov-Razumikhin method for impulsive functional differential equations and applications to the population dynamics,” Journal of Computational and Applied Mathematics, vol. 130, no. 1-2, pp. 163–171, 2001.
• Z. G. Luo and J. H. Shen, “New Razumikhin type theorems for impulsive functional differential equations,” Applied Mathematics and Computation, vol. 125, no. 2-3, pp. 375–386, 2002.
• Q. Wang and X. Liu, “Impulsive stabilization of delay differential systems via the Lyapunov-Razumikhin method,” Applied Mathematics Letters, vol. 20, no. 8, pp. 839–845, 2007.
• W. H. Chen and W. X. Zheng, “Robust stability and ${H}_{\infty }$-control of uncertain impulsive systems with time-delay,” Automatica, vol. 45, no. 1, pp. 109–117, 2009.
• X. L. Fu and X. D. Li, “Razumikhin-type theorems on exponential stability of impulsive infinite delay differential systems,” Journal of Computational and Applied Mathematics, vol. 224, no. 1, pp. 1–10, 2009.
• X. D. Li, “New results on global exponential stabilization of impulsive functional differential equations with infinite delays or finite delays,” Nonlinear Analysis–-Real World Applications, vol. 11, no. 5, pp. 4194–4201, 2010.
• F. L. Lian, J. Moyne, and D. Tilbury, “Modelling and optimal controller design of networked control systems with multiple delays,” International Journal of Control, vol. 76, no. 6, pp. 591–606, 2003.
• A. Khadra, X. Z. Liu, and X. M. Shen, “Analyzing the robustness of impulsive synchronization coupled by linear delayed impulses,” IEEE Transactions on Automatic Control, vol. 54, no. 4, pp. 923–928, 2009.
• Y. Zhang and J. T. Sun, “Stability of impulsive functional differential equations,” Nonlinear Analysis–-Theory, Methods & Applications, vol. 68, no. 12, pp. 3665–3678, 2008.
• P. Cheng, Z. Wu, and L. L. Wang, “New results on global exponential stability of impulsive functional differential systems with delayed impulses,” Abstract and Applied Analysis, vol. 2012, Article ID 376464, 13 pages, 2012.
• D. W. Lin, X. D. Li, and D. O'Regan, “Stability analysis of generalized impulsive functional differential equations,” Mathematical and Computer Modelling, vol. 55, no. 5-6, pp. 1682–1690, 2012.
• W. H. Chen and W. X. Zheng, “Exponential stability of nonlinear time-delay systems with delayed impulse effects,” Automatica, vol. 47, no. 5, pp. 1075–1083, 2011.
• X. R. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, UK, 1997.
• A. M. Samoilenko and O. M. Stanzhytskiy, Qualitative and Asymptotic Analisis of Differential Equations with Random Peryurbations, vol. 78 of World Scientific Series on Nonlinear Science, Series A, 2011.
• S. G. Peng and B. G. Jia, “Some criteria on pth moment stability of impulsive stochastic functional differential equations,” Statistics & Probability Letters, vol. 80, no. 13-14, pp. 1085–1092, 2010.
• P. Cheng and F. Q. Deng, “Global exponential stability of impulsive stochastic functional differential systems,” Statistics & Probability Letters, vol. 80, no. 23-24, pp. 1854–1862, 2010.
• C. X. Li, J. T. Sun, and R. Y. Sun, “Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects,” Journal of the Franklin Institute, vol. 347, no. 7, pp. 1186–1198, 2010.
• S. G. Peng and Y. Zhang, “Razumikhin-type theorems on pth moment exponential stability of impulsivestochastic delay differential equations,” IEEE Transactions on Automatic Control, vol. 55, no. 8, pp. 1085–1092, 2010.
• L. J. Pan and J. D. Cao, “Exponential stability of impulsive stochastic functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 382, no. 2, pp. 672–685, 2011.
• F. Q. Yao and F. Q. Deng, “Exponential stability in terms of two measures of impulsive stochastic functional differential systems via comparison principle,” Statistics & Probability Letters, vol. 82, no. 6, pp. 1151–1159, 2012.
• F. Q. Yao and F. Q. Deng, “Stability of impulsive stochastic functional differential systems in terms of two measures via comparison approach,” Science China Information Sciences, vol. 55, no. 6, pp. 1313–1322, 2012.
• J. Liu, X. Z. Liu, and W. C. Xie, “Impulsive stabilization of stochastic functional differential equations,” Applied Mathematics Letters, vol. 24, no. 3, pp. 264–269, 2011.
• F. Q. Yao, F. Q. Deng, and P. Cheng, “Exponential stability of impulsive stochastic functional differential systems with delayed impulses,” Abstract and Applied Analysis, vol. 2013, Article ID 548712, 8 pages, 2013.
• M. S. Alwan, X. Z. Liu, and W. C. Xie, “Existence, continuation, and uniqueness problems of stochastic impulsive systems with time delay,” Journal of the Franklin Institute, vol. 347, no. 7, pp. 1317–1333, 2010. \endinput