##
**Delay-dependent exponential stability for impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms.**
*(English)*
Zbl 1221.35440

Summary: A class of impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion is formulated and investigated. By employing delay differential inequality and the linear matrix inequality (LMI) optimization approach, some sufficient conditions ensuring global exponential stability of equilibrium point for impulsive Cohen-Grossberg neural networks with time-varying delays and diffusion are obtained. In particular, the estimate of the exponential convergence rate is also provided, which depends on system parameters, diffusion effect and impulsive disturbed intention. It is believed that these results are significant and useful for the design and applications of Cohen-Grossberg neural networks. An example is given to show the effectiveness of the results obtained here.

### MSC:

35R12 | Impulsive partial differential equations |

35B35 | Stability in context of PDEs |

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

### Keywords:

Cohen-Grossberg neural networks; impulses; delays; global exponential stability; linear matrix inequality
PDFBibTeX
XMLCite

\textit{X. Zhang} et al., Commun. Nonlinear Sci. Numer. Simul. 16, No. 3, 1524--1532 (2011; Zbl 1221.35440)

Full Text:
DOI

### References:

[1] | Cohen, M.; Grossberg, S., Absolute stability and global pattern formation and parallel memory storage by competitive neural networks, IEEE Trans Syst Man Cybern, 13, 815-826 (1983) · Zbl 0553.92009 |

[2] | Cao, J.; Liang, J., Boundedness and stability for Cohen-Grossberg neural network with time-varying delays, J Math Anal Appl, 296, 665-685 (2004) · Zbl 1044.92001 |

[3] | Yuan, K.; Cao, J., An analysis of global asymptotic stability of delayed Cohen-Grossberg neural networks via non-smooth analysis, IEEE Trans Circuits Syst I, 52, 9, 1854-1861 (2005) · Zbl 1374.34291 |

[4] | Arik, S.; Orman, Z., Global stability analysis of Cohen-Grossberg neural networks with time varying delays, Phys Lett A, 341, 410-421 (2005) · Zbl 1171.37337 |

[5] | Zhang, J.; Suda, Y.; Komine, H., Global exponential stability of Cohen-Grossberg neural networks with variable delays, Phys Lett A, 338, 44-55 (2005) · Zbl 1136.34347 |

[6] | Liao, X.; Li, C., Global attractivity of Cohen-Grossberg model with finite and infinite delays, J Math Anal Appl, 315, 244-262 (2006) · Zbl 1098.34062 |

[7] | Jiang, M.; Shen, Y.; Liao, X., Boundedness and global exponential stability for generalized Cohen-Grossberg neural networks with variable delay, Appl Math Comp, 172, 379-393 (2006) · Zbl 1090.92004 |

[8] | Song, Q.; Cao, J., Stability analysis of Cohen-Grossberg neural network with both time-varying and continuously distributed delays, J Comp Appl Math, 197, 188-203 (2006) · Zbl 1108.34060 |

[9] | Yang, Z.; Xu, D., Impulsive effects on stability of Cohen-Grossberg neural networks with variable delays, Appl Math Comput, 177, 63-78 (2006) · Zbl 1103.34067 |

[10] | Li, K., Stability analysis for impulsive Cohen-Grossberg neural networks with time-varying delays and distributed delays, Nonlinear Anal Real World Appl, 10, 2784-2798 (2009) · Zbl 1162.92002 |

[11] | Chua, L. O., Passivity and complexity, IEEE Trans Circ Syst I Fundam Theory Appl, 46, 71-82 (1999) · Zbl 0948.92002 |

[12] | Itoh, M.; Chua, L. O., Complexity of reaction-diffusion CNN, Int J Bifurc Chaos, 16, 2499-2527 (2006) · Zbl 1185.37191 |

[13] | Wang, L.; Xu, D., Global exponential stability of Hopfield reaction-diffusion neural networks with variable delays, Sci China Ser F, 46, 466-474 (2003) · Zbl 1186.82062 |

[14] | Li, K.; Song, Q., Exponential stability of impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms, Neurocomputing, 72, 231-240 (2008) |

[15] | Li, Z.; Li, K., Stability analysis of impulsive Cohen-Grossberg neural networks with distributed delays and reaction-diffusion terms, Appl Math Model, 33, 1337-1348 (2009) · Zbl 1168.35382 |

[16] | Li, K., Global exponential stability of impulsive fuzzy cellular neural networks with delays and diffusion, Int J Bifurc Chaos, 19, 245-261 (2009) · Zbl 1170.35326 |

[17] | Qiu, J., Exponential stability of impulsive neural networks with time-varying delays and reaction-diffusion terms, Neurocomputing, 70, 1102-1108 (2007) |

[18] | Qiu, J.; Cao, J., Delay-dependent exponential stability for a class of neural networks with time delays and reaction-diffusion terms, J Franklin Inst, 346, 301-314 (2009) · Zbl 1166.93368 |

[19] | Liang, J.; Cao, J., Global exponential stability of reaction-diffusion recurrent neural networks with time-varying delays, Phys Lett A, 314, 434-442 (2003) · Zbl 1052.82023 |

[20] | Chen, A.; Huang, L.; Cao, J., Exponential stability of delayed bidirectional associative memory neural networks with reaction diffusion terms, Int J Syst Sci, 38, 421-432 (2007) · Zbl 1130.35348 |

[21] | Lou, X.; Cui, B., New criteria on global exponential stability of BAM neural networks with distributed delays and reaction-diffusion terms, Int J Neural Syst, 17, 43-52 (2007) |

[22] | Lou, X.; Cui, B., Asymptotic synchronization of a class of neural networks with reaction-diffusion terms and time-varying delays, Comput Math Appl, 52, 897-904 (2006) · Zbl 1126.35083 |

[23] | Cui, B.; Lou, X., Global asymptotic stability of BAM neural networks with distributed delays and reaction-diffusion terms, Chaos Solitons Fractals, 27, 1347-1354 (2006) · Zbl 1084.68095 |

[24] | Zhao, H.; Wang, G., Existence of periodic oscillatory solution of reaction-diffusion neural networks with delays, Phys Lett A, 343, 372-383 (2005) · Zbl 1194.35221 |

[25] | Song, Q.; Cao, J., Dynamics of bidirectional associative memory networks with distributed delays and reaction-diffusion terms, Nonlinear Anal Real World Appl, 8, 345-361 (2007) · Zbl 1114.35103 |

[26] | Lu, J. G., Global exponential stability and periodicity of reaction-diffusion delayed recurrent neural networks with Dirichlet boundary conditions, Chaos Solitons Fractals, 35, 116-125 (2008) · Zbl 1134.35066 |

[27] | Wang, J.; Lu, J. G., Global exponential stability of fuzzy cellular neural networks with delays and reaction-diffusion terms, Chaos Solitons Fractals, 38, 878-885 (2008) · Zbl 1146.35315 |

[28] | Lu, J. G.; Lu, L. J., Global exponential stability and periodicity of reaction-diffusion recurrent neural networks with distributed delays and Dirichlet boundary conditions, Chaos Solitons Fractals, 39, 1538-1549 (2009) · Zbl 1197.35144 |

[29] | Xu, S.; Chen, T.; Lam, J., Robust \(H_∞\) filtering for uncertain Markovian jumps systems with mode-dependent time delays, IEEE Trans Autom Control, l48, 900-907 (2003) · Zbl 1364.93816 |

[30] | Yue, D.; Xu, S. F.; Liu, Y. Q., Differential inequality with delay and impulse and its applications to design robust control, Control Theory Appl, 16, 4, 519-524 (1999) · Zbl 0995.93063 |

[31] | Cao, J.; Wang, J., Global asymptotic stability of a general class of recurrent neural networks with time delays, IEEE Trans Circuit Syst I, 50, 1, 34-44 (2003) · Zbl 1368.34084 |

[32] | Pan, J.; Liu, X.; Zhong, S., Stability criteria for impulsive reaction-diffusion Cohen-Grossberg neural networks with time-varying delays, Math Comput Model, 51, 1037-1050 (2010) · Zbl 1198.35033 |

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.