×

Stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks. (English) Zbl 1193.34168

Summary: We investigate the stochastic functional differential equations with infinite delay. Some sufficient conditions are derived to ensure the \(p\)th moment exponential stability and \(p\)th moment global asymptotic stability of stochastic functional differential equations with infinite delay by using Razumikhin method and Lyapunov functions. Based on the obtained results, we further study the \(p\)th moment exponential stability of stochastic recurrent neural networks with unbounded distributed delays. The result extends and improves the earlier publications. Two examples are given to illustrate the applicability of the obtained results.

MSC:

34K50 Stochastic functional-differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Mohammed, S., Stochastic Functional Differential Equations (1984), Pitman · Zbl 0584.60066
[2] Mao, X., Stochastic Differential Equations and Applications (1997), Horwood · Zbl 0874.60050
[3] Øksendal, B., Stochastic Differential Equations: An Introduction with Applications (1995), Springer-Verlag · Zbl 0841.60037
[4] Friedman, A., Stochastic Differential Equations and Applications (1975), Academic Press: Academic Press New York
[5] Taniguchi, T., Successive approximations to solutions of stochastic differential equations, J. Differential Equations, 96, 152-169 (1992) · Zbl 0744.34052
[6] 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
[7] Shen, Y.; Luo, Q.; Mao, X., The improved LaSalle-type theorems for stochastic functional differential equations, J. Math. Anal. Appl., 318, 134-154 (2006) · Zbl 1090.60059
[8] Liu, K., Uniform stability of autonomous linear stochastic functional differential equations in infinite dimensions, Stochastic Process. Appl., 115, 1131-1165 (2005) · Zbl 1075.60078
[9] Halidias, N.; Ren, Y., An existence theorem for stochastic functional differential equations with delays under weak assumptions, Stat. Probab. Lett., 78, 2864-2867 (2008) · Zbl 1156.60047
[10] 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
[11] Xu, D.; Yang, Z.; Huang, Y., Existence-uniqueness and continuation theorems for stochastic functional differential equations, J. Differential Equations, 245, 1681-1703 (2008) · Zbl 1161.34055
[12] Chang, M., On Razumikhin-type stability conditions for stochastic functional differential equations, Math. Modelling, 5, 299-307 (1984) · Zbl 0574.60065
[13] Mao, X., Razumikhin-type theorems on exponential stability of stochastic functional differential equations, Stochastic Process. Appl., 65, 233-250 (1996) · Zbl 0889.60062
[15] Luo, J., Stability of stochastic partial differential equations with infinite delays, J. Comput. Appl. Math., 222, 364-371 (2008) · Zbl 1151.60336
[16] Wei, F.; Wang, K., The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay, J. Math. Anal. Appl., 331, 516-531 (2007) · Zbl 1121.60064
[17] Ren, Y.; Lua, S.; Xia, N., Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay, J. Comput. Appl. Math., 220, 364-372 (2008) · Zbl 1152.34388
[18] Fu, X.; Li, X., Razumikhin-type theorems on exponential stability of impulsive infinite delay differential systems, J. Comput. Appl. Math., 224, 1-10 (2009) · Zbl 1179.34079
[19] Liu, Y.; Ge, W., Stability theorems and existence results for periodic solutions of nonlinear impulsive delay differential equations with variable coefficients, Nonlinear Anal., 57, 363-399 (2004) · Zbl 1064.34051
[20] Shen, J.; Yan, J., Razumikhin type stability theorems for impulsive functional differential equations, Nonlinear Anal., 33, 519-531 (1998) · Zbl 0933.34083
[21] Fu, X.; Yan, B.; Liu, Y., Introduction of Impulsive Differential Systems (2005), Science Press: Science Press Beijing
[22] Li, X., Uniform asymptotic stability and global stability of impulsive infinite delay differential equations, Nonlinear Anal., 70, 1975-1983 (2009) · Zbl 1175.34094
[23] Gopalsamy, K.; Zhang, B., On delay differential equations with impulses, J. Math. Anal. Appl., 139, 110-122 (1989) · Zbl 0687.34065
[24] Rakkiyappan, R.; Balasubramaniam, P., LMI conditions for stability of stochastic recurrent neural networks with distributed delays, Chaos Solitons Fractals, 40, 1688-1696 (2009) · Zbl 1198.34161
[25] Rakkiyappan, P.; Balasubramaniam, R., Global asymptotic stability of stochastic recurrent neural networks with multiple discrete delays and unbounded distributed delays, Appl. Math. Comput., 204, 680-686 (2008) · Zbl 1152.93049
[26] Huang, C.; He, Y.; Huang, L.; Zhu, W., \(p\) th moment stability analysis of stochastic recurrent neural networks with time-varying delays, Inform. Sci., 178, 2194-2203 (2008) · Zbl 1144.93030
[27] Sun, Y.; Cao, J., \(p\) th moment exponential stability of stochastic recurrent neural networks with time-varying delays, Nonlinear Anal., 8, 1171-1185 (2007) · Zbl 1196.60125
[28] Zhang, G.; Lin, Y., Functional Analysis (1987), Peking University Press: Peking University Press Beijing
[29] Beckenbach, E.; Bellman, R., Inequalities (1961), Springer-Verlag: Springer-Verlag New York · Zbl 0186.09606
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.