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**Exponential stability for a class of stochastic reaction-diffusion Hopfield neural networks with delays.**
*(English)*
Zbl 1244.93121

Summary: This paper studies the asymptotic behavior for a class of delayed reaction-diffusion Hopfield neural networks driven by finite-dimensional Wiener processes. Some new sufficient conditions are established to guarantee the mean square exponential stability of this system by using Poincaré’s inequality and stochastic analysis technique. The proof of the almost surely exponential stability for this system is carried out by using the Burkholder-Davis-Gundy inequality, the Chebyshev inequality and the Borel-Cantelli lemma. Finally, an example is given to illustrate the effectiveness of the proposed approach, and the simulation is also given by using the Matlab.

### MSC:

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

93E20 | Optimal stochastic control |

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

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

### Software:

Matlab
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\textit{X. Liang} and \textit{L. Wang}, J. Appl. Math. 2012, Article ID 693163, 12 p. (2012; Zbl 1244.93121)

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### References:

[1] | X. X. Liao and Y. L. Gao, “Stability of Hopfield neural networks with reactiondiffusion terms,” Acta Electronica Sinica, vol. 28, pp. 78-82, 2000. |

[2] | L. S. 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 |

[3] | L. S. Wang and Y. F. Wang, “Stochastic exponential stability of the delayed reaction diffusion interval neural networks with Markovian jumpling parameters,” Physics Letters A, vol. 356, pp. 346-352, 2008. |

[4] | J. K. Hale and V. S. M. Lunel, Introduction to Functional-Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer, Berlin, Germany, 1993. · Zbl 0787.34002 |

[5] | S.-E. A. Mohammed, Stochastic Functional Differential Equations, vol. 99 of Research Notes in Mathematics, Pitman, London, UK, 1984. · Zbl 0584.60066 |

[6] | L. S. Wang and Y. Y. Gao, “Global exponential robust stability of reactiondiffusion interval neural networks with time varying delays,” Physics Letters A, vol. 305, pp. 343-348, 2006. · Zbl 1195.35179 |

[7] | L. S. Wang, R. Zhang, and Y. Wang, “Global exponential stability of reaction-diffusion cellular neural networks with S-type distributed time delays,” Nonlinear Analysis: Real World Applications, vol. 10, no. 2, pp. 1101-1113, 2009. · Zbl 1167.35404 |

[8] | H. Y. Zhao and G. L. Wang, “Existence of periodic oscillatory solution of reactiondiffusion neural networks with delays,” Physics Letters A, vol. 343, pp. 372-382, 2005. · Zbl 1194.35221 |

[9] | J. G. Lu and L. J. Lu, “Global exponential stability and periodicity of reaction-diffusion recurrent neural networks with distributed delays and Dirichlet boundary conditions,” Chaos, Solitons and Fractals, vol. 39, no. 4, pp. 1538-1549, 2009. · Zbl 1197.35144 |

[10] | X. Mao, Stochastic Differential Equations and Applications, Horwood, 1997. · Zbl 0892.60057 |

[11] | J. Sun and L. Wan, “Convergence dynamics of stochastic reaction-diffusion recurrent neural networks with delays,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 15, no. 7, pp. 2131-2144, 2005. · Zbl 1092.35530 |

[12] | M. Itoh and L. O. Chua, “Complexity of reaction-diffusion CNN,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 16, no. 9, pp. 2499-2527, 2006. · Zbl 1185.37191 |

[13] | X. Lou and B. Cui, “New criteria on global exponential stability of BAM neural networks with distributed delays and reaction-diffusion terms,” International Journal of Neural Systems, vol. 17, pp. 43-52, 2007. · Zbl 05153443 |

[14] | Q. Song and Z. Wang, “Dynamical behaviors of fuzzy reaction-diffusion periodic cellular neural networks with variable coefficients and delays,” Applied Mathematical Modelling, vol. 33, no. 9, pp. 3533-3545, 2009. · Zbl 1185.35129 |

[15] | Q. Song, J. Cao, and Z. Zhao, “Periodic solutions and its exponential stability of reaction-diffusion reccurent neural networks with distributed time delays,” Nonlinear Analysis B, vol. 8, pp. 345-361, 2007. · Zbl 1114.35103 |

[16] | K. Liu, “Lyapunov functionals and asymptotic stability of stochastic delay evolution equations,” Stochastics and Stochastics Reports, vol. 63, no. 1-2, pp. 1-26, 1998. · Zbl 0947.93037 |

[17] | R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1988. · Zbl 0662.35001 |

[18] | G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, vol. 44 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, UK, 1992. · Zbl 0761.60052 |

[19] | I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta Kharkov, 2002. · Zbl 1100.37047 |

[20] | R. Jahanipur, “Stochastic functional evolution equations with monotone nonlinearity: existence and stability of the mild solutions,” Journal of Differential Equations, vol. 248, no. 5, pp. 1230-1255, 2010. · Zbl 1210.34112 |

[21] | T. Taniguchi, “Almost sure exponential stability for stochastic partial functional-differential equations,” Stochastic Analysis and Applications, vol. 16, no. 5, pp. 965-975, 1998. · Zbl 0911.60054 |

[22] | T. Caraballo, K. Liu, and A. Truman, “Stochastic functional partial differential equations: existence, uniqueness and asymptotic decay property,” Proceedings the Royal Society of London A, vol. 456, no. 1999, pp. 1775-1802, 2000. · Zbl 0972.60053 |

[23] | M. Kamrani and S. M. Hosseini, “The role of coefficients of a general SPDE on the stability and convergence of a finite difference method,” Journal of Computational and Applied Mathematics, vol. 234, no. 5, pp. 1426-1434, 2010. · Zbl 1201.65012 |

[24] | D. J. Higham, “Mean-square and asymptotic stability of the stochastic theta method,” SIAM Journal on Numerical Analysis, vol. 38, no. 3, pp. 753-769, 2000. · Zbl 0982.60051 |

[25] | P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, vol. 23, Springer, Berlin, Germany, 3rd edition, 1999. · Zbl 0752.60043 |

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