RAZUMIKHIN-TYPE THEOREM FOR NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY

doi:10.1016/S0252-9602(11)60312-3

Abstract

Abstract:

This paper establishes the Razumikhin-type theorem on stability for neutral stochastic functional differential equations with unbounded delay. To overcome difficulties from unbounded delay, we develop several different techniques to investigate stability. To show our idea clearly, we examine neutral stochastic delay differential equations with un-bounded delay and linear neutral stochastic Volterra unbounded-delay-integro-differential equations.

Key words: neutral stochastic functional differential equations, Razumikhin-type theo-rem, stability, exponential stability, polynomial stability

CLC Number:

34K40

WU Fu-Ke, HU Shi-Geng, MAO Xue-Rong. RAZUMIKHIN-TYPE THEOREM FOR NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY[J].Acta mathematica scientia,Series B, 2011, 31(4): 1245-1258.

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References

[1] Blythe S, Mao X, Shah A. Razumikhin-type theorems on stability of stochastic neural networks with delays. Stoch Anal Appl, 2001, 19(1): 85–101

[2] Hale J K, Lunel S M V. Introduction to Functional Differential Equations. Springer-Verlag, 1993

[3] He X. The Lyapunov functionals for delay Lotka–Volterra-type models. SIAM J Appl Math, 1998, 58: 1222–1236.

[4] Hu Y, Wu F, Huang C. Robustness of exponential stability of a class of stochastic functional differential equations with infinite delay. Automatica, 2009, 45: 2577–2584

[5] Iserles A, Liu Y. On neutral functional-differential equations with proportional delays. J Math Anal Appl, 1997, 207: 73–95

[6] Kolmanovskii V B, Nosov V R. Stability of Functional Differential Equations. Academic Press, 1986

[7] Kolmanovskii V B, Myshkis A. Applied Theory of Functional Differential Equations. Kluwer Academic Publishers

[8] Mao X. Exponential stability in mean square of neutral stochastic differential functional equations. Systems & Control Letters, 1995, 26: 245–251

[9] Mao X. Razumikhin-type theorems on exponential stability of stochastic functional differential equations. Stoch Proc Appl, 1996, 65: 233–250

[10] Mao X. Stochatic Differential Equations and Applications. Horwood: Chichester, 1997

[11] Mao X. Razumikhin-type theorems on exponential stability of neutral stochastic functional differential. SIAM J Math Anal, 1997, 28: 389–401

[12] Mao X. Comments on “an improved Razumikhin-type theorem and its applications”. IEEE Trans Aut Contr, 1997, 43(3): 429–430

[13] Mao X, Rodkina A, Koroleva N. Razumikhin-type theorems for neutral stochastic functional differential equations. Funct Differ Equ, 1998, 5(1/2): 195–211

[14] Mao X, Lam J, Xu S, Gao H. Razumikhin method and exponential stability of hybrid stochastic delay interval systems. J Math Anal Appl, 2006, 314: 45–66

[15] Randielovic J, Jankovic S. On the pth moment exponential stability criteria of neutral stochastic functional differential equations. J Math Anal Appl, 2007, 326: 266–280

[16] Razumikhin B S. On the stability of systems with a delay. Rossiiskaya Akademiya Nauk: Prikladnaya Matematika i Mekhanika, 1956, 20: 500–512

[17] Razumikhin B S. Application of Lyapunov’s method to problems in the stability of systems with a delay. Rossiiskaya Akademiya Nauk: Avtomatika i Telemekhanika, 1960, 21: 740–749

[18] Wu F, Xu Y. Stochastic Lotka-Volterra population dynamics with infinite delay. SIAM J Appl Math, 2009, 70: 641–657

[19] Wu F, Hu S, Huang C. Robustness of general decay stability of nonlinear neutral stochastic functional differential equations with infinite delay. System & Control Letter, 2010, 59: 95–202