##
**Asymptotic stability of impulsive reaction-diffusion cellular neural networks with time-varying delays.**
*(English)*
Zbl 1235.93197

Summary: This work addresses the asymptotic stability for a class of impulsive cellular neural networks with time-varying delays and reaction-diffusion. By using the impulsive integral inequality of Gronwall-Bellman type and Hardy-Sobolev inequality as well as piecewise continuous Lyapunov functions, we summarize some new and concise sufficient conditions ensuring the global exponential asymptotic stability of the equilibrium point. The provided stability criteria are applicable to Dirichlet boundary condition and showed to be dependent on all of the reaction-diffusion coefficients, the dimension of the space, the delay, and the boundary of the spatial variables. Two examples are finally illustrated to demonstrate the effectiveness of our obtained results.

### MSC:

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

35K57 | Reaction-diffusion equations |

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

PDF
BibTeX
XML
Cite

\textit{Y. Zhang}, J. Appl. Math. 2012, Article ID 501891, 17 p. (2012; Zbl 1235.93197)

Full Text:
DOI

### References:

[1] | L. O. Chua and L. Yang, “Cellular neural networks: theory,” IEEE Transactions on Circuits and Systems, vol. 35, no. 10, pp. 1257-1272, 1988. · Zbl 0663.94022 |

[2] | L. O. Chua and L. Yang, “Cellular neural networks: applications,” IEEE Transactions on Circuits and Systems, vol. 35, no. 10, pp. 1273-1290, 1988. · Zbl 0663.94022 |

[3] | J. Cao, “New results concerning exponential stability and periodic solutions of delayed cellular neural networks,” Physics Letters A, vol. 307, no. 2-3, pp. 136-147, 2003. · Zbl 1006.68107 |

[4] | J. Cao, “On stability of cellular neural networks with delay,” IEEE Transactions on Circuits and Systems, vol. 40, no. 3, pp. 157-165, 1993. · Zbl 0792.68115 |

[5] | P. P. Civalleri and M. Gilli, “A set of stability criteria for delayed cellular neural networks,” IEEE Transactions on Circuits and Systems. I, vol. 48, no. 4, pp. 494-498, 2001. · Zbl 0792.68115 |

[6] | J. J. Hopfield, “Neurons with graded response have collective computational properties like those of two-state neurons,” Proceedings of the National Academy of Sciences of the United States of America, vol. 81, no. 10 I, pp. 3088-3092, 1984. · Zbl 1371.92015 |

[7] | J. Yan and J. Shen, “Impulsive stabilization of functional-differential equations by Lyapunov-Razumikhin functions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 37, no. 2, pp. 245-255, 1999. · Zbl 0994.82066 |

[8] | X. Liu and Q. Wang, “Impulsive stabilization of high-order Hopfield-type neural networks with time-varying delays,” IEEE Transactions on Neural Networks, vol. 19, no. 1, pp. 71-79, 2008. |

[9] | X. Liu, “Stability results for impulsive differential systems with applications to population growth models,” Dynamics and Stability of Systems, vol. 9, no. 2, pp. 163-174, 1994. · Zbl 0951.34049 |

[10] | S. Arik and V. Tavsanoglu, “On the global asymptotic stability of delayed cellular neural networks,” IEEE Transactions on Circuits and Systems. I, vol. 47, no. 4, pp. 571-574, 2000. · Zbl 0808.34056 |

[11] | L. O. Chua and T. Roska, “Stability of a class of nonreciprocal cellular neural networks,” IEEE Transactions on Circuits and Systems, vol. 37, no. 12, pp. 1520-1527, 1990. |

[12] | Z.-H. Guan and G. Chen, “On delayed impulsive Hopfield neural networks,” Neural Networks, vol. 12, no. 2, pp. 273-280, 1999. |

[13] | Q. Zhang, X. Wei, and J. Xu, “On global exponential stability of delayed cellular neural networks with time-varying delays,” Applied Mathematics and Computation, vol. 162, no. 2, pp. 679-686, 2005. · Zbl 0997.90095 |

[14] | D. D. Bainov and P. S. Simeonov, Systems with Impulse Effect, Ellis Horwood Ltd., Chichester, UK, 1989. · Zbl 1114.34337 |

[15] | I. M. Stamova, Stability Analysis of Impulsive Functional Differential Equations, Walter de Gruyter, Berlin, Germany, 2009. · Zbl 1189.34001 |

[16] | M. A. Arbib, Brains, Machines, and Mathematics, Springer, New York, NY, USA, 2nd edition, 1987. · Zbl 0645.68001 |

[17] | S. Haykin, Neural Networks: A Comprehensive Foundation, Prentice-Hall, Englewood Cliffs, NJ, USA, 1998. · Zbl 0828.68103 |

[18] | H. Ak\cca, R. Alassar, V. Covachev, Z. Covacheva, and E. Al-Zahrani, “Continuous-time additive Hopfield-type neural networks with impulses,” Journal of Mathematical Analysis and Applications, vol. 290, no. 2, pp. 436-451, 2004. · Zbl 1057.68083 |

[19] | G. T. Stamov, “Almost periodic models of impulsive Hopfield neural networks,” Journal of Mathematics of Kyoto University, vol. 49, no. 1, pp. 57-67, 2009. · Zbl 1058.34007 |

[20] | G. T. Stamov and I. M. Stamova, “Almost periodic solutions for impulsive neural networks with delay,” Applied Mathematical Modelling, vol. 31, no. 7, pp. 1263-1270, 2007. · Zbl 1136.34332 |

[21] | S. Ahmad and I. M. Stamova, “Global exponential stability for impulsive cellular neural networks with time-varying delays,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 3, pp. 786-795, 2008. · Zbl 1177.34058 |

[22] | X. Liu, K. L. Teo, and B. Xu, “Exponential stability of impulsive high-order Hopfield-type neural networks with time-varying delays,” IEEE Transactions on Neural Networks, vol. 16, no. 6, pp. 1329-1339, 2005. |

[23] | J. Qiu, “Exponential stability of impulsive neural networks with time-varying delays and reaction-diffusion terms,” Neurocomputing, vol. 70, no. 4-6, pp. 1102-1108, 2007. |

[24] | K. Li and Q. Song, “Exponential stability of impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms,” Neurocomputing, vol. 72, no. 1-3, pp. 231-240, 2008. · Zbl 05718819 |

[25] | J. Pan, X. Liu, and S. Zhong, “Stability criteria for impulsive reaction-diffusion Cohen-Grossberg neural networks with time-varying delays,” Mathematical and Computer Modelling, vol. 51, no. 9-10, pp. 1037-1050, 2010. · Zbl 1151.34061 |

[26] | V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6, World Scientific, Teaneck, NJ, USA, 1989. · Zbl 1198.35033 |

[27] | Adimurthi, “Hardy-Sobolev inequality in H1(\Omega ) and its applications,” Communications in Contemporary Mathematics, vol. 4, no. 3, pp. 409-434, 2002. · Zbl 1005.35072 |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.