##
**Stability of the stochastic reaction-diffusion neural network with time-varying delays and \(p\)-Laplacian.**
*(English)*
Zbl 1263.92004

Summary: The main aim of this paper is to discuss moment exponential stability for a stochastic reaction-diffusion neural network with time-varying delays and p-Laplacian. Using the Itô formula, a delay differential inequality and the characteristics of the neural network, and the algebraic conditions for the moment exponential stability of the non-constant equilibrium solution are derived. An example is also given for illustration.

### MSC:

92B20 | Neural networks for/in biological studies, artificial life and related topics |

35K57 | Reaction-diffusion equations |

35R60 | PDEs with randomness, stochastic partial differential equations |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

PDFBibTeX
XMLCite

\textit{P. Qingfei} et al., J. Appl. Math. 2012, Article ID 405939, 10 p. (2012; Zbl 1263.92004)

Full Text:
DOI

### References:

[1] | X. Liao and J. Wang, “Global dissipativity of continuous-time recurrent neural networks with time delay,” Physical Review E, vol. 68, no. 1, Article ID 016118, 6 pages, 2003. · doi:10.1103/PhysRevE.68.016118 |

[2] | L. Wang and D. Xu, “Stability for Hopfield neural networks with time delay,” Journal of Vibration and Control, vol. 8, no. 1, pp. 13-18, 2002. · Zbl 1012.93054 · doi:10.1177/1077546302008001527 |

[3] | D. Xu, H. Zhao, and H. Zhu, “Global dynamics of Hopfield neural networks involving variable delays,” Computers & Mathematics with Applications, vol. 42, no. 1-2, pp. 39-45, 2001. · Zbl 0990.34036 · doi:10.1016/S0898-1221(01)00128-6 |

[4] | L. S. Wang and D. Y. Xu, “Stability analysis of Hopfield neural networks with time delay,” Applied Mathematics and Mechanics, vol. 23, no. 1, pp. 65-70, 2002. · Zbl 1015.92004 · doi:10.1007/BF02437731 |

[5] | X. Liao, X. Mao, J. Wang, and Z. Zeng, “Algebraic conditions of stability for Hopfield neural network,” Science in China F, vol. 47, no. 1, pp. 113-125, 2004. · Zbl 1186.82060 · doi:10.1360/02yf0206 |

[6] | X. X. Liao, S. Z. Yang, S. J. Cheng, and Y. Shen, “Stability of general neural networks with reaction-diffusion,” Sciena in China E, vol. 32, no. 1, pp. 87-94, 2002 (Chinese). |

[7] | L. Wang and D. Xu, “Global exponential stability of Hopfield reaction-diffusion neural networks with time-varying delays,” Science in China F, vol. 46, no. 6, pp. 466-474, 2003. · Zbl 1186.82062 · doi:10.1360/02yf0146 |

[8] | S. Blythe, X. Mao, and X. Liao, “Stability of stochastic delay neural networks,” Journal of the Franklin Institute, vol. 338, no. 4, pp. 481-495, 2001. · Zbl 0991.93120 · doi:10.1016/S0016-0032(01)00016-3 |

[9] | S. Hu, X. Liao, and X. Mao, “Stochastic Hopfield neural networks,” Journal of Physics A, vol. 36, pp. 2235-2249, 2003. · Zbl 1042.82036 · doi:10.1088/0305-4470/36/9/303 |

[10] | J. R. Niu, Z. F. Zhang, and D. Y. Xu, “Exponential stability in mean square of a stochastic Cohen-Grossberg neural network with time-varying delays,” Chinese Journal of Engineering Mathematics, vol. 22, no. 6, pp. 1001-1005, 2005 (Chinese). |

[11] | Y. Shen, M. H. Jiang, and H. S. Yao, “Exponential stability of cellular neural networks,” Acta Mathematica Scientia A, vol. 25, no. 2, pp. 264-268, 2005 (Chinese). · Zbl 1259.93089 |

[12] | Q. Luo, F. Deng, J. Bao, B. Zhao, and Y. Fu, “Stabilization of stochastic Hopfield neural network with distributed parameters,” Science in China F, vol. 47, no. 6, pp. 752-762, 2004. · Zbl 1187.35128 · doi:10.1360/03yf0332 |

[13] | Z. Zifang, The stability of uncertain dynamical systems and applications [M.S. thesis], Sichuan University, 2004. · Zbl 1088.45503 |

[14] | S. J. Long and L. Xiang, “A nonlinear measure method for stability of Hopfield neural networks with time-varying delays,” Journal of Sichuan University, vol. 44, no. 2, pp. 249-252, 2007 (Chinese). · Zbl 1164.34532 |

[15] | Z.-F. Zhang and D.-Y. Xu, “A note on stability of stochastic delay neural networks,” Chinese Journal of Engineering Mathematics, vol. 27, no. 4, pp. 720-730, 2010. · Zbl 1240.93362 |

[16] | Z. F. Zhang and J. Deng, “Stability of stochastic reaction-diffusion Hopfield neural network with time-varying delays,” Journal of Sichuan University, vol. 47, no. 3, pp. 251-256, 2010. · Zbl 1240.60192 · doi:10.3969/j.issn.0490-6756.2010.03.008 |

[17] | D. Xu, “Stability criteria of large-scale systems of the neutral type,” in Proceedings of the12th IMACS World Congress on Scientific Computations, R. Vichnevetsky, P. Borne, and J. Vignes, Eds., vol. 1, p. 213, Gerfidn-cite Scientifique, Pairs, France, 1988. |

[18] | W. Rudin, Real and Complex Analysis, McGraw-Hill, 1974. · Zbl 0278.26001 |

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.