# zbMATH — the first resource for mathematics

Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects. (English) Zbl 1207.34104
From the text: We discuss a class of stochastic differential delay equations with nonlinear impulsive effects of the form
$\begin{cases} dy(t) =\{-a_1(t)y(t)-a_2(t)y(t-\tau(t))\}\,dt\\ \qquad\;+\{-b_1(t)y(t)-b_2(t)y(t-\tau(t))]\,dw(t),\quad & t\neq t_k,\\ y(t_k^+)-y(t_k)=I_k(y(t_k)), & t=t_k,\;k\in\mathbb N,\end{cases}$
where $$I_k\in C(\mathbb R,\mathbb R)$$, $$k\in\mathbb N$$ are continuous functions with $$I_k(0)\equiv 0$$.
The purpose of this paper is to build a bridge between the given stochastic impulsive delay equation and a corresponding stochastic delay equation without impulsive effects, and to establish some stability criteria for these systems. Furthermore, the desired conditions are given explicitly.

##### MSC:
 34K50 Stochastic functional-differential equations 34K20 Stability theory of functional-differential equations 34K45 Functional-differential equations with impulses
Full Text:
##### References:
 [1] Shen, J.H., Razumikhin techniques in impulsive functional differential equation, Nonlinear anal., 36, 119-130, (1999) · Zbl 0939.34071 [2] Benbouziane, Z.; Boucherif, A.; Bouguima, S.M., Existence result for impulsive third order periodic boundary value problems, Appl. math. comput., 206, 728-737, (2008) · Zbl 1159.34021 [3] Luo, Z.G.; Nieto, J.J., New results for the periodic boundary value problem for impulsive integro-differential equations, Nonlinear anal., 70, 2248-2260, (2009) · Zbl 1166.45002 [4] Liu, X.Z.; Ballinger, G., Uniform asymptotic stability of impulsive delay differential equations, Comput. math. appl., 41, 903-915, (2001) · Zbl 0989.34061 [5] Yan, J.R., Stability for impulsive delay differential equations, Nonlinear anal., 63, 66-80, (2005) · Zbl 1082.34069 [6] Zhang, Y.; Sun, J.T., Stability of impulsive infinite delay differential equations, Appl. math. lett., 19, 1100-1106, (2006) · Zbl 1125.34345 [7] Song, Q.K.; Cao, J.D., Dynamical behaviors of discrete-time fuzzy cellular neural networks with variable delays and impulses, J. franklin inst., 345, 39-59, (2008) · Zbl 1167.93369 [8] Sun, J.T.; Han, Q.L.; Jiang, X.F., Impulsive control of time-delay systems using delayed impulse and its application to impulsive master – slave synchronization, Phys. lett. A, 372, 6375-6380, (2008) · Zbl 1225.94018 [9] Zhang, J.; Gui, Z.J., Periodic solutions of nonautonomous cellular neural networks with impulse and delays, Nonlinear anal., 10, 1891-1903, (2009) · Zbl 1160.92005 [10] Ding, W.; Han, M.A.; Li, M.L., Exponential lag synchronization of delayed fuzzy cellular neural networks with impulses, Phys. lett. A, 373, 832-837, (2009) · Zbl 1228.34075 [11] Wu, H.J.; Sun, J.T., p-moment stability of stochastic differential equations with impulsive jump and Markovian switching, Automatica, 42, 1753-1759, (2006) · Zbl 1114.93092 [12] Liu, B., Stability of solutions of stochastic impulsive systems via comparison approach, IEEE trans. automat. control, 53, 2128-2133, (2008) · Zbl 1367.93523 [13] Zhang, H.; Guan, Z.H.; Feng, G., Reliable dissipative control for stochastic impulsive systems, Automatica, 44, 1004-1010, (2008) · Zbl 1283.93258 [14] Mao, X.R.; Selfridge, C., Stability of stochastic interval systems with time delays, Systems control lett., 42, 279-290, (2001) · Zbl 0974.93047 [15] Blythe, S.B.; Mao, X.R.; Liao, X.X., Stability of stochastic delay neural networks, J. franklin inst., 338, 481-495, (2001) · Zbl 0991.93120 [16] Lu, C.-Y.; Su, T.-J.; Tsai, J.S.-H., On robust stabilization of uncertain stochastic time-delay systems an LMI-based approach, J. franklin inst., 342, 473-487, (2005) · Zbl 1091.93035 [17] Luo, J.W., A note on exponential stability in pth Mean of solutions of stochastic delay differential equations, J. comput. appl. math., 198, 143-148, (2007) · Zbl 1110.65009 [18] Wan, L.; Zhou, Q.H., Exponential stability of stochastic reaction – diffusion cohen – grossberg neural networks with delays, Appl. math. comput., 206, 818-824, (2008) · Zbl 1255.60109 [19] Li, X.L.; Cao, J.D., Adaptive synchronization for delayed neural networks with stochastic perturbation, J. franklin inst., 345, 779-791, (2008) · Zbl 1169.93350 [20] Muthukumar, P.; Balasubramaniam, P., Approximate controllability of nonlinear stochastic evolution systems with time-varying delays, J. franklin inst., 346, 65-80, (2009) · Zbl 1298.93070 [21] Zhang, J.H.; Shi, P.; Qiu, J.Q., Non-fragile guaranteed cost control for uncertain stochastic nonlinear time-delay systems, J. franklin inst., 346, 676-690, (2009) · Zbl 1298.93364 [22] Yang, J.; Zhong, S.M.; Luo, W.P., Mean square stability analysis of impulsive stochastic differential equations with delays, J. comput. appl. math., 216, 474-483, (2008) · Zbl 1142.93035 [23] Zhang, H.; Guan, Z.H., Stability analysis on uncertain stochastic impulsive systems with time-delay, Phys. lett. A, 372, 6053-6059, (2008) · Zbl 1223.37014 [24] Xu, L.G.; Xu, D.Y., Mean square exponential stability of impulsive control stochastic systems with time-varying delay, Phys. lett. A, 373, 328-333, (2009) · Zbl 1227.34082 [25] Zhao, S.W.; Sun, J.T.; Wu, H.J., Stability of linear stochastic differential delay systems under impulsive control, IET control theory appl., 3, 1547-1552, (2009) [26] Sakthivel, R.; Luo, J., Asymptotic stability of impulsive stochastic partial differential equations with infinite delays, J. math. anal. appl., 356, 1-6, (2009) · Zbl 1166.60037 [27] Li, C.X.; Sun, J.T., Stability analysis of nonlinear stochastic differential delay systems under impulsive control, Phys. lett. A, 374, 1154-1158, (2010) · Zbl 1248.90030
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.