Exponential stability of impulsive stochastic functional differential systems with delayed impulses. (English) Zbl 1271.93169

Summary: A class of generalized impulsive stochastic functional differential systems with delayed impulses is considered. By employing piecewise continuous Lyapunov functions and the Razumikhin techniques, several criteria on the exponential stability and uniform stability in terms of two measures for the mentioned systems are obtained, which show that unstable stochastic functional differential systems may be stabilized by appropriate delayed impulses. Based on the stability results, delayed impulsive controllers which mean square exponentially stabilize linear stochastic delay systems are proposed. Finally, numerical examples are given to verify the effectiveness and advantages of our results.


93E15 Stochastic stability in control theory
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
93D30 Lyapunov and storage functions
Full Text: DOI


[1] Luo, Z.; Shen, J., Impulsive stabilization of functional differential equations with infinite delays, Applied Mathematics Letters, 16, 5, 695-701, (2003) · Zbl 1068.93054
[2] Liu, X., Stability of impulsive control systems with time delay, Mathematical and Computer Modelling, 39, 4-5, 511-519, (2004) · Zbl 1081.93021
[3] Liu, X.; Wang, Q., The method of Lyapunov functionals and exponential stability of impulsive systems with time delay, Nonlinear Analysis: Theory, Methods & Applications, 66, 7, 1465-1484, (2007) · Zbl 1123.34065
[4] Liu, X.; Wang, Q., On stability in terms of two measures for impulsive systems of functional differential equations, Journal of Mathematical Analysis and Applications, 326, 1, 252-265, (2007) · Zbl 1112.34059
[5] Wang, Q.; Liu, X., Impulsive stabilization of delay differential systems via the Lyapunov-Razumikhin method, Applied Mathematics Letters, 20, 8, 839-845, (2007) · Zbl 1159.34347
[6] Li, X., New results on global exponential stabilization of impulsive functional differential equations with infinite delays or finite delays, Nonlinear Analysis: Real World Applications, 11, 5, 4194-4201, (2010) · Zbl 1210.34103
[7] Fu, X.; Li, X., Razumikhin-type theorems on exponential stability of impulsive infinite delay differential systems, Journal of Computational and Applied Mathematics, 224, 1, 1-10, (2009) · Zbl 1179.34079
[8] Chen, W.-H.; Zheng, W. X., Robust stability and \(H_\infty\)-control of uncertain impulsive systems with time-delay, Automatica, 45, 1, 109-117, (2009) · Zbl 1154.93406
[9] Zhang, Y.; Sun, J., Stability of impulsive functional differential equations, Nonlinear Analysis: Theory, Methods & Applications, 68, 12, 3665-3678, (2008) · Zbl 1152.34053
[10] Lin, D.; Li, X.; O’Regan, D., Stability analysis of generalized impulsive functional differential equations, Mathematical and Computer Modelling, 55, 5-6, 1682-1690, (2012) · Zbl 1255.34077
[11] Wu, Q.; Zhou, J.; Xiang, L., Global exponential stability of impulsive differential equations with any time delays, Applied Mathematics Letters, 23, 2, 143-147, (2010) · Zbl 1210.34105
[12] Khadra, A.; Liu, X. Z.; Shen, X., Analyzing the robustness of impulsive synchronization coupled by linear delayed impulses, Institute of Electrical and Electronics Engineers, 54, 4, 923-928, (2009) · Zbl 1367.34084
[13] Liu, J.; Liu, X.; Xie, W.-C., Impulsive stabilization of stochastic functional differential equations, Applied Mathematics Letters, 24, 3, 264-269, (2011) · Zbl 1209.34097
[14] Cheng, P.; Deng, F.; Peng, Y., Robust exponential stability and delayed-state-feedback stabilization of uncertain impulsive stochastic systems with time-varying delay, Communications in Nonlinear Science and Numerical Simulation, 17, 12, 4740-4752, (2012) · Zbl 1263.93232
[15] Yang, T.; Chua, L. O., Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication, IEEE Transactions on Circuits and Systems. I, 44, 10, 976-988, (1997)
[16] Liu, X., Impulsive stabilization and control of chaotic system, Nonlinear Analysis: Theory, Methods & Applications, 47, 1081-1092, (2001) · Zbl 1042.93523
[17] Li, C.; Liao, X.; Yang, X.; Huang, T., Impulsive stabilization and synchronization of a class of chaotic delay systems, Chaos, 15, 4, (2005) · Zbl 1144.37371
[18] Liu, X.; Rohlf, K., Impulsive control of a Lotka-Volterra system, IMA Journal of Mathematical Control and Information, 15, 3, 269-284, (1998) · Zbl 0949.93069
[19] Liu, X.; Willms, A. R., Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft, Mathematical Problems in Engineering, 277-299, (1996) · Zbl 0876.93014
[20] Carter, T. E., Optimal impulsive space trajectories based on linear equations, Journal of Optimization Theory and Applications, 70, 2, 277-297, (1991) · Zbl 0732.49025
[21] Yu-Jun, N.; Wei, X.; Hong-Wu, L., Asymptotical p-moment stability of stochastic impulsive differential equations and its application in impulsive control, Communications in Theoretical Physics, 53, 1, 110-114, (2010) · Zbl 1225.93117
[22] Li, C.; Chen, L.; Aihara, K., Impulsive control of stochastic systems with applications in chaos control, chaos synchronization, and neural networks, Chaos, 18, 2, (2008) · Zbl 1307.93378
[23] Huang, L.; Mao, X., Robust delayed-state-feedback stabilization of uncertain stochastic systems, Automatica, 45, 5, 1332-1339, (2009) · Zbl 1162.93389
[24] Alwan, M. S.; Liu, X.; Xie, W.-C., Existence, continuation, and uniqueness problems of stochastic impulsive systems with time delay, Journal of the Franklin Institute, 347, 7, 1317-1333, (2010) · Zbl 1205.60107
[25] Peng, S.; Zhang, Y., Razumikhin-type theorems on \(p\) th moment exponential stability of impulsive stochastic delay differential equations, Institute of Electrical and Electronics Engineers, 55, 8, 1917-1922, (2010) · Zbl 1368.93771
[26] Lakshmikantham, V.; Liu, X. Z., Stability Analysis in Terms of Two Measures, (1993), River Edge, NJ, USA: World Scientific, River Edge, NJ, USA · Zbl 0797.34056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.