# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Exponential stability of impulsive stochastic functional differential equations. (English) Zbl 1222.60043
Authors’ abstract: “We investigate the $p$-th moment and almost sure exponential stability of impulsive stochastic differential equations with finite delay by using Lyapunov method. Several stability theorems of impulsive stochastic functional differential equations with finite delay are derived. These new results are employed to impulsive stochastic equations with bounded time-varying delays and stochastically perturbed equations. Also, an example and simulations are given to show that impulsive effects play an important role in $p$-th moment and almost sure exponential stability of stochastic functional differential equations with finite delay.”

##### MSC:
 60H10 Stochastic ordinary differential equations 93E15 Stochastic stability
##### Keywords:
stochastic functional differential equations
Full Text:
##### References:
 [1] Hale, J. K.; Lunel, S. M. V.: Introduction to functional differential equations, (1993) · Zbl 0787.34002 [2] Mao, X.: Exponential stability of stochastic differential equations, (1994) · Zbl 0806.60044 [3] Mao, X.: Stochastic differential equations and applications, (1997) · Zbl 0892.60057 [4] Chang, M.: On razumikhin-type stability conditions for stochastic functional differential equations, Math. modelling 5, 299-307 (1984) · Zbl 0574.60065 · doi:10.1016/0270-0255(84)90007-1 [5] Karatzas, I.; Shreve, S. E.: Brownian motion and stochastic calculus, (1991) · Zbl 0734.60060 [6] Mao, X.: Razumikhin type theorems on exponential stability of neutral stochastic functional differential equations, SIAM J. Math. anal. 28, No. 2, 389-401 (1997) · Zbl 0876.60047 · doi:10.1137/S0036141095290835 [7] Mao, X.: Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Stochastic process. Appl. 65, 233-250 (1996) · Zbl 0889.60062 · doi:10.1016/S0304-4149(96)00109-3 [8] Xu, D. Y.; Yang, Z. G.; Huang, Y. M.: Existence-uniqueness and continuation theorems for stochastic functional differential equations, J. differential equations 245, 1681-1703 (2008) · Zbl 1161.34055 · doi:10.1016/j.jde.2008.03.029 [9] Liu, B.; Liu, X.; Teo, K.; Wang, Q.: Razumikhin-type theorems on exponential stability of impulsive delay systems, IMA J. Appl. math. 71, 47-61 (2006) · Zbl 1128.34047 · doi:10.1093/imamat/hxh091 [10] Anokhin, A.; Berezansky, L.; Braverman, E.: Exponential stability of linear delay impulsive differential equations, J. math. Anal. appl. 193, 923-941 (1995) · Zbl 0837.34076 · doi:10.1006/jmaa.1995.1275 [11] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations, (1989) · Zbl 0718.34011 [12] Samoilenko, A. M.; Perestyuk, N. A.: Impulsive differential equations, (1995) · Zbl 0837.34003 [13] Shen, J.; Yan, J.: Razumikhin type stability theorems for impulsive functional differential equations, Nonlinear anal. 33, 519-531 (1998) · Zbl 0933.34083 · doi:10.1016/S0362-546X(97)00565-8 [14] Luo, J. W.: Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays, J. math. Anal. appl. 342, 753-760 (2008) · Zbl 1157.60065 · doi:10.1016/j.jmaa.2007.11.019 [15] Liu, K.; Truman, A.: A note on almost sure exponential stability for stochastic partial functional differential equations, Statist. probab. Lett. 50, 273-278 (2000) · Zbl 0966.60059 · doi:10.1016/S0167-7152(00)00103-6 [16] Liu, B.: Stability of solutions for stochastic impulsive systems via comparison approach, IEEE trans. Automat. control 53, 2128-2133 (2008) [17] Janković, S.; Randjelović, J.; Jovanović, M.: Razumikhin-type exponential stability criteria of neutral stochastic functional differential equations, J. math. Anal. appl. 355, 811-820 (2009) · Zbl 1166.60040 · doi:10.1016/j.jmaa.2009.02.011 [18] Peng, S. G.; Jia, B. G.: Some criteria on pth moment stability of impulsive stochastic functional differential equations, Statist. probab. Lett. 80, 1085-1092 (2010) · Zbl 1197.60056 · doi:10.1016/j.spl.2010.03.002 [19] Cheng, P.; Deng, F. Q.: Global exponential stability of impulsive stochastic functional differential systems, Statist. probab. Lett. 80, 1854-1862 (2010) · Zbl 1205.60110 · doi:10.1016/j.spl.2010.08.011 [20] 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 · doi:10.1016/j.jmaa.2009.02.002 [21] 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 [22] Liu, X. Z.; Wang, Q.: The method of Lyapunov functionals and exponential stability of impulsive systems with time delay, Nonlinear anal. 66, No. 7, 1465-1484 (2007) · Zbl 1123.34065 · doi:10.1016/j.na.2006.02.004 [23] Wang, Q.; Liu, X. Z.: Impulsive stabilization of delay differential systems via the Lyapunov-razumikhin method, Appl. math. Lett. 20, No. 8, 839-845 (2007) · Zbl 1159.34347 · doi:10.1016/j.aml.2006.08.016 [24] Taniguchi, T.; Liu, K.; Truman, A.: Existence, uniqueness, and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces, J. differential equations 181, 72-91 (2002) · Zbl 1009.34074 · doi:10.1006/jdeq.2001.4073 [25] Anguraj, A.; Vinodkumar, A.: Existence, uniqueness and stability results of random impulsive semilinear differential systems, Nonlinear anal. Hybrid syst. 4, 475-483 (2010) · Zbl 1208.34093 · doi:10.1016/j.nahs.2009.11.004 [26] Wu, Q. J.; Zhou, J.; Xiang, L.: Global exponential stability of impulsive differential equations with any time delays, Appl. math. Lett. 23, 143-147 (2010) · Zbl 1210.34105 · doi:10.1016/j.aml.2009.09.001