Stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks

Journal of Computational and Applied Mathematics

Volumn 234, Issue 2, 2010, Pages 407-417

  Li, Xiaodi ^a   Fu, Xilin ^b  

^a XIAMEN UNIVERSITY   (China)

^b SHANDONG NORMAL UNIVERSITY   (China)

Author keywords

Infinite delay; Lyapunov functions; Razumikhin method; Recurrent neural networks; Stability; Stochastic functional differential equations; Unbounded distributed delays

Indexed keywords

GLOBAL ASYMPTOTIC STABILITY; INFINITE DELAY; PTH MOMENT EXPONENTIAL STABILITY; RAZUMIKHIN METHOD; STABILITY ANALYSIS; STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATION; STOCHASTIC RECURRENT NEURAL NETWORK; SUFFICIENT CONDITIONS; UNBOUNDED DISTRIBUTED DELAYS;

ASYMPTOTIC STABILITY; DIFFERENTIAL EQUATIONS; EXPONENTIAL FUNCTIONS; LYAPUNOV FUNCTIONS; STOCHASTIC SYSTEMS;

RECURRENT NEURAL NETWORKS;

EID: 77649274176     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2009.12.033     Document Type: Article

References (29)

1. Mohammed S. Stochastic Functional Differential Equations (1984), Pitman
2. Mao X. Stochastic Differential Equations and Applications (1997), Horwood
3. Øksendal B. Stochastic Differential Equations: An Introduction with Applications. fourth ed. (1995), Springer-Verlag
4. Friedman A. Stochastic Differential Equations and Applications (1975), Academic Press, New York
5. Taniguchi T. Successive approximations to solutions of stochastic differential equations. J. Differential Equations 96 (1992) 152-169
6. Taniguchi T., Liu K., and Truman A. Existence, uniqueness, and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces. J. Differential Equations 181 (2002) 72-91
7. Shen Y., Luo Q., and Mao X. The improved LaSalle-type theorems for stochastic functional differential equations. J. Math. Anal. Appl. 318 (2006) 134-154
8. Liu K. Uniform stability of autonomous linear stochastic functional differential equations in infinite dimensions. Stochastic Process. Appl. 115 (2005) 1131-1165
9. Halidias N., and Ren Y. An existence theorem for stochastic functional differential equations with delays under weak assumptions. Stat. Probab. Lett. 78 (2008) 2864-2867
10. Taniguchi T., Liu K., and Truman A. Existence, uniqueness, and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces. J. Differential Equations 181 (2002) 72-91
11. Xu D., Yang Z., and Huang Y. Existence-uniqueness and continuation theorems for stochastic functional differential equations. J. Differential Equations 245 (2008) 1681-1703
12. Chang M. On Razumikhin-type stability conditions for stochastic functional differential equations. Math. Modelling 5 (1984) 299-307
13. Mao X. Razumikhin-type theorems on exponential stability of stochastic functional differential equations. Stochastic Process. Appl. 65 (1996) 233-250
14. Y. Ren, N. Xia, Existence, uniqueness and stability of the solutions to neutral stochastic functional differential equations with infinite delay, Appl. Math. Comput. (in press)
15. Luo J. Stability of stochastic partial differential equations with infinite delays. J. Comput. Appl. Math. 222 (2008) 364-371
16. Wei F., and Wang K. The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay. J. Math. Anal. Appl. 331 (2007) 516-531
17. Ren Y., Lua S., and Xia N. Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay. J. Comput. Appl. Math. 220 (2008) 364-372
18. Fu X., and Li X. Razumikhin-type theorems on exponential stability of impulsive infinite delay differential systems. J. Comput. Appl. Math. 224 (2009) 1-10
19. Liu Y., and Ge W. Stability theorems and existence results for periodic solutions of nonlinear impulsive delay differential equations with variable coefficients. Nonlinear Anal. 57 (2004) 363-399
20. Shen J., and Yan J. Razumikhin type stability theorems for impulsive functional differential equations. Nonlinear Anal. 33 (1998) 519-531
21. Fu X., Yan B., and Liu Y. Introduction of Impulsive Differential Systems (2005), Science Press, Beijing
22. Li X. Uniform asymptotic stability and global stability of impulsive infinite delay differential equations. Nonlinear Anal. 70 (2009) 1975-1983
23. Gopalsamy K., and Zhang B. On delay differential equations with impulses. J. Math. Anal. Appl. 139 (1989) 110-122
24. Rakkiyappan R., and Balasubramaniam P. LMI conditions for stability of stochastic recurrent neural networks with distributed delays. Chaos Solitons Fractals 40 (2009) 1688-1696
25. Rakkiyappan P., and Balasubramaniam R. Global asymptotic stability of stochastic recurrent neural networks with multiple discrete delays and unbounded distributed delays. Appl. Math. Comput. 204 (2008) 680-686
26. Huang C., He Y., Huang L., and Zhu W. pth moment stability analysis of stochastic recurrent neural networks with time-varying delays. Inform. Sci. 178 (2008) 2194-2203
27. Sun Y., and Cao J. pth moment exponential stability of stochastic recurrent neural networks with time-varying delays. Nonlinear Anal. 8 (2007) 1171-1185
28. Zhang G., and Lin Y. Functional Analysis (1987), Peking University Press, Beijing
29. Beckenbach E., and Bellman R. Inequalities (1961), Springer-Verlag, New York