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**On global stability criterion of neural networks with continuously distributed delays.**
*(English)*
Zbl 1141.93054

Summary: Based on the Lyapunov’s second method and the Linear Matrix Inequality (LMI) optimization approach, this paper presents a new sufficient condition for global asymptotic stability of the equilibrium point for a class of neural networks with discrete and distributed delays. The stability condition is expressed in terms of LMIs, which can be solved easily by various convex optimization algorithms. A numerical example is given to show the less conservatism and effectiveness of proposed method.

### MSC:

93D20 | Asymptotic stability in control theory |

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

90C25 | Convex programming |

93C15 | Control/observation systems governed by ordinary differential equations |

34D23 | Global stability of solutions to ordinary differential equations |

### Software:

LMI toolbox
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\textit{J. H. Park}, Chaos Solitons Fractals 37, No. 2, 444--449 (2008; Zbl 1141.93054)

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### References:

[1] | Chua, L.O.; Yang, L., Cellular neural networks: theory and applications, IEEE trans circuits systems I, 35, 1257-1290, (1988) · Zbl 0663.94022 |

[2] | Ramesh, M.; Narayanan, S., Chaos control of bonhoeffer-van der Pol oscillator using neural networks, Chaos, solitons & fractals, 12, 2395-2405, (2001) · Zbl 1004.37067 |

[3] | Chen, C.J.; Liao, T.L.; Hwang, C.C., Exponential synchronization of a class of chaotic neural networks, Chaos, solitons & fractals, 24, 197-206, (2005) · Zbl 1060.93519 |

[4] | Cannas, B.; Cincotti, S.; Marchesi, M.; Pilo, F., Learning of chua’s circuit attractors by locally recurrent neural networks, Chaos, solitons & fractals, 12, 2109-2115, (2001) · Zbl 0981.68135 |

[5] | Cao, J.; Ho, D.W.C., A general framework for global asymptotic stability analysis of delayed neural networks based on LMI approach, Chaos, solitons & fractals, 24, 1317-1329, (2005) · Zbl 1072.92004 |

[6] | Zhou, S.; Liao, X.; Wu, Z.; Wong, K., Hopf bifurcation in a control system for the washout filter-based delayed neural equation, Chaos, solitons & fractals, 23, 101-115, (2005) · Zbl 1071.93042 |

[7] | Park, J.H.; Jung, H.Y., On the design of nonfragile guaranteed cost controller for a class of uncertain dynamic systems with state delays, Appl math comput, 150, 245-257, (2004) · Zbl 1036.93054 |

[8] | Cao, J., Global asymptotic stability of neural networks with transmission delays, Int J syst sci, 31, 1313-1316, (2000) · Zbl 1080.93517 |

[9] | Park, J.H., Design of dynamic controller for neutral differential systems with delay in control input, Chaos, solitons & fractals, 23, 503-509, (2005) · Zbl 1144.34386 |

[10] | Arik, S., Global robust stability analysis of neural networks with discrete time delays, Chaos, solitons & fractals, 26, 1407-1414, (2005) · Zbl 1122.93397 |

[11] | Park, J.H.; Kwon, O., Controlling uncertain neutral dynamic systems with delay in control input, Chaos, solitons & fractals, 26, 805-812, (2005) · Zbl 1091.34045 |

[12] | Arik, S., An analysis of global asymptotic stability of delayed cellular neural networks, IEEE trans neural netw, 13, 1239-1242, (2002) |

[13] | Park, J.H., A novel criterion for global asymptotic stability of BAM neural networks with time delays, Chaos, solitons, & fractals, 29, 446-453, (2006) · Zbl 1121.92006 |

[14] | Park, J.H., Robust stability of bidirectional associative memory neural networks with time delays, Phys lett A, 349, 494-499, (2006) |

[15] | Singh, V., Robust stability of cellular neural networks with delay: linear matrix inequality approach, IEE proc control theory appl, 151, 125-129, (2004) |

[16] | Zhao, H., Global asymptotic stability of Hopfield neural network involving distributed delays, Neural networks, 17, 47-53, (2004) · Zbl 1082.68100 |

[17] | Liang, J.; Cao, J., Global asymptotic stability of bi-directional associative memory networks with distributed delays, Appl math comput, 152, 415-424, (2004) · Zbl 1046.94020 |

[18] | Zhao, H., Global stability of bidirectional associative memory neural networks with distributed delays, Phys lett A, 297, 182-190, (2002) · Zbl 0995.92002 |

[19] | Wang, Z.; Liu, Y.; Liu, X., On global asymptotic stability of neural networks with discrete and distributed delays, Phys lett A, 345, 299-308, (2005) · Zbl 1345.92017 |

[20] | Yang, H.; Chu, T., LMI conditions for stability of neural networks with distributed delays, Chaos, solitons & fractals, 34, 2, 557-563, (2007) · Zbl 1146.34331 |

[21] | Boyd, B.; Ghaoui, L.E.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in systems and control theory, (1994), SIAM Philadelphia |

[22] | Gahinet, P.; Nemirovski, A.; Laub, A.; Chilali, M., LMI control toolbox user’s guide, (1995), The Mathworks Massachusetts |

[23] | Hale, J.; Verduyn Lunel, S.M., Introduction to functional differential equations, (1993), Springer-Verlag New York · Zbl 0787.34002 |

[24] | Moon, Y.S.; Park, P.; Kwon, W.H.; Lee, Y.S., Int J control, 74, 1447-1455, (2001) |

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