Robust exponential stability and delayed-state-feedback stabilization of uncertain impulsive stochastic systems with time-varying delay. (English) Zbl 1263.93232

Summary: This paper studies the problem of robust exponential stability and delayed-state-feedback stabilization of uncertain impulsive stochastic systems with time-varying delay. The state variables on the impulses are assumed dependent on the present state variables as well as delayed state variables. Based on the Razumikhin techniques and Lyapunov functions, some robust mean-square exponential stability criteria are derived in terms of linear matrix inequalities. The results show that the system is stable if the impulses’ frequency and amplitude are suitably related to the increase or decrease of the continuous flows. Furthermore, robust delayed-state-feedback controllers that mean-square exponentially stabilize the uncertain impulsive stochastic systems are proposed. Finally, several numerical examples are given to show the effectiveness of the results.


93E15 Stochastic stability in control theory
93D09 Robust stability
93D15 Stabilization of systems by feedback
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Full Text: DOI


[1] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of impulsive differential equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002
[2] Haddad, W. M.; Chellaboina, V.; Nersesov, S. G., Impulsive and hybrid dynamical systems: stability, dissipativity, and control (2006), Princeton University Press: Princeton University Press Princeton · Zbl 1114.34001
[3] Yang, T.; Chua, L. O., Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication, IEEE Trans Circuits Syst, 44, 10, 976-988 (1997)
[4] Lian, F.; Moyne, J.; Tilbury, D., Modelling and optimal controller design of networked control systems with multiple delays, INT Control, 76, 6, 591-606 (2003) · Zbl 1050.93038
[5] Khadra, A.; Liu, X. Z.; Shen, X. M., Analyzing the robustness of impulsive synchronization coupled by linear delayed impulses, IEEE Trans Automat Control, 54, 4, 923-928 (2009) · Zbl 1367.34084
[6] Haykin, S., Neural networks (1994), Prentice-Hall: Prentice-Hall NJ · Zbl 0828.68103
[7] Niculescu, S., Delay effects on stability: a robust control approach (2001), Springer: Springer New York · Zbl 0997.93001
[8] Liu, X. Z.; Ballinger, G., Uniform asymptotic stability of impulsive delay differential equations, Comput Math Appl, 41, 7, 903-915 (2001) · Zbl 0989.34061
[9] Liu, X. Z.; Wang, Q., The method of Lyapunov functionals and exponential stability of impulsive systems with time delay, Nonlinear Anal, 66, 7, 1465-1484 (2007) · Zbl 1123.34065
[10] Yang, Z. C.; Xu, D. Y., Stability analysis and design of impulsive control systems with time delay, IEEE Trans Automat Control, 52, 8, 1448-1454 (2007) · Zbl 1366.93276
[11] Zhang, Y.; Sun, J. T., Stability of impulsive functional differential equations, Nonlinear Anal, 68, 12, 3665-3678 (2008) · Zbl 1152.34053
[12] Wu, Q. J.; Zhou, J.; Xiang, L., Global exponential stability of impulsive differential equations with any time delays, Appl Math Lett, 23, 2, 143-147 (2010) · Zbl 1210.34105
[13] Li, X. D., New results on global exponential stabilization of impulsive functional differential equations with infinite delays or finite delays, Nonlinear Anal RWA, 11, 5, 4194-4201 (2010) · Zbl 1210.34103
[14] Balasubramaniam, P.; Vembarasan, V., Global asymptotic stability results for BAM neural networks of neutral-type with time delays in the leakage term under impulsive perturbations, INT Comp Math, 88, 15, 3271-3291 (2011) · Zbl 1247.34122
[15] XR, Mao, Stochastic differential equations and applications (1997), Horwood Publishing Limited, Horwood: Horwood Publishing Limited, Horwood England · Zbl 0892.60057
[16] Yang, Z. G.; Xu, D. Y.; Xang, L., Exponential p-stability of impulsive stochastic differential equations with delays, Phys Lett A, 359, 2, 129-137 (2006) · Zbl 1236.60061
[17] Chen, W. H.; Wang, J. G.; Tang, Y. J.; Lu, X. M., Robust \(H_\infty\) control of uncertain linear impulsive stochastic systems, INT Robust Nonlinear, 18, 13, 1348-1371 (2008) · Zbl 1298.93346
[18] Sakthivel, R.; Luo, J. W., Asymptotic stability of nonlinear impulsive stochastic differential equations, Stat Probabil Lett, 79, 9, 1219-1223 (2009) · Zbl 1166.60316
[19] Xu, L. G.; Xu, D. Y., Mean square exponential stability of impulsive control stochastic systems with time-varying delay, Phys Lett A, 373, 3, 328-333 (2009) · Zbl 1227.34082
[20] Cheng, P.; Deng, F. Q., Global exponential stability of impulsive stochastic functional differential systems, Stat Probabil Lett, 80, 23-24, 1854-1862 (2010) · Zbl 1205.60110
[21] Li, C. X.; Sun, J. T.; Sun, R. Y., Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects, J Franklin Inst, 347, 7, 1186-1198 (2010) · Zbl 1207.34104
[22] Liu, J.; Liu, X. Z.; Xie, W. C., Impulsive stabilization of stochastic functional differential equations, Appl Math Lett, 24, 3, 264-269 (2011) · Zbl 1209.34097
[23] Liu, Y. J.; Liao, D.; Wang, P., Stochastic asymptotical stability for stochastic impulsive differential equations and it is application to chaos synchronization, Commun Nonlinear Sci Numer Simulat, 17, 2, 505-512 (2012) · Zbl 1246.34055
[24] Mao, X., Robustness of exponential stability of stochastic differential delay equations, IEEE Trans Automat Control, 41, 3, 442-447 (1996) · Zbl 0851.93074
[25] Xu, S. Y.; Chen, T. W., Robust \(H_\infty\) control for uncertain stochastic systems with state delay, IEEE Trans Automat Control, 47, 12, 2089-2094 (2002) · Zbl 1364.93755
[26] Yue, D.; Han, Q. L., Delay-dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and Markovian switching, IEEE Trans Automat Control, 50, 2, 217-222 (2005) · Zbl 1365.93377
[27] Chen, W. H.; Guan, Z. H.; Lu, X. M., Delay-dependent exponential stability of uncertain stochastic systems with multiple delays: an LMI approach, Syst Control Lett, 54, 6, 547-555 (2005) · Zbl 1129.93547
[28] Gao, H. J.; James, L.; Wang, C. H., Robust energy-to-peak filter design for stochastic time-delay systems, Syst Control Lett, 55, 2, 101-111 (2006) · Zbl 1129.93538
[29] Balasubramaniam, P.; Vembarasan, V.; Rakkiyappan, R., Delay-dependent robust exponential state estimation of Markovian jumping fuzzy Hopfield neural networks with mixed random time-varying delays, Commun Nonlinear Sci Numer Simulat, 16, 4, 2109-2129 (2011) · Zbl 1221.93254
[30] Balasubramaniam, P.; Vembarasan, V.; Rakkiyappan, R., Delay-dependent robust asymptotic state estimation of Takagi-Sugeno fuzzy Hopfield neural networks with mixed interval time-varying delays, Expert Syst Appl, 39, 1, 472-481 (2012)
[31] Liu, B.; Teo, K. L.; Liu, X. Z., Robust exponential stabilization for large-scale uncertain impulsive systems with coupling time-delays, Nonlinear Anal, 68, 5, 1169-1183 (2008) · Zbl 1154.34041
[32] Zong, G. D.; Xu, S. Y.; Wu, Y. Q., Robust \(H_\infty\) stabilization for uncertain switched impulsive control systems with state delay: an LMI approach, Nonlinear Anal Hybrid Syst, 2, 4, 1287-1300 (2008) · Zbl 1163.93386
[33] 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
[34] Balasubramaniam, P.; Vembarasan, V., Robust stability of uncertain fuzzy BAM neural networks of neutral-type with Markovian jumping parameters and impulses, Comput Math Appl, 62, 4, 1838-1861 (2011) · Zbl 1231.34139
[35] Zhang, Y., Robust exponential stability of uncertain impulsive delay difference equations with continuous time, J Franklin Inst, 348, 8, 1965-1982 (2011) · Zbl 1239.39014
[36] Liu, X.; Zhong, S. M.; Ding, X. Y., Robust exponential stability of impulsive switched systems with switching delays: a Razumikhin approach, Commun Nonlinear Sci Numer Simulat, 17, 4, 1805-1812 (2012) · Zbl 1239.93104
[37] Mao, X. R.; Lam, J.; Huang, L. R., Stabilisation of hybrid stochastic differential equations by delay feedback control, Syst Control Lett, 57, 11, 927-935 (2008) · Zbl 1149.93027
[38] Huang, L. R.; Mao, X. R., Robust delayed-state-feedback stabilization of uncertain stochastic systems, Automatica, 45, 5, 1332-1339 (2009) · Zbl 1162.93389
[39] Alwan, M. S.; Liu, X. Z.; Xie, W. C., Existence, continuation,and uniqueness problems of stochastic impulsive systems with time delay, J Franklin Inst, 347, 7, 1317-1333 (2010) · Zbl 1205.60107
[40] Wang, Y.; Xie, L.; de Souza, C. E., Robust control of a class of uncertain nonlinear systems, Syst Control Lett, 19, 2, 139-149 (1992) · Zbl 0765.93015
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.