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**Exponential stability in the mean square for stochastic neural networks with mixed time-delays and Markovian jumping parameters.**
*(English)*
Zbl 1176.92007

Summary: The stability analysis problem is considered for a class of stochastic neural networks with mixed time-delays and Markovian jump parameters. The mixed delays include discrete and distributed time-delays, and the jump parameters are generated from a continuous-time discrete-state homogeneous Markov process. The aim of this paper is to establish some criteria under which delayed stochastic neural networks are exponentially stable in the mean square. By constructing suitable Lyapunov functionals, several stability conditions are derived on the basis of inequality techniques and stochastic analysis. An example is also provided in the end of this paper to demonstrate the usefulness of the proposed criteria.

### MSC:

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

60J20 | Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) |

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

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\textit{G. Wang} et al., Nonlinear Dyn. 57, No. 1--2, 209--218 (2009; Zbl 1176.92007)

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### References:

[1] | Cao, J.: On stability of delayed cellular neural networks. Phys. Lett. A 261, 303–308 (1999) · Zbl 0935.68086 |

[2] | Cao, J., Wang, J.: Global asymptotic stability of a general class of recurrent neural networks with time-varying delays. IEEE Trans. Circ. Syst. I 50, 34–44 (2003) · Zbl 1368.34084 |

[3] | 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 |

[4] | Arik, S.: An analysis of exponential stability of delayed neural networks with time varying delays. Neural Netw. 17, 1027–1031 (2004) · Zbl 1068.68118 |

[5] | Xu, S., Lam, J., Ho, D.W.C.: Novel global asymptotical stability criteria for delayed cellular neural networks. IEEE Trans. Circ. Syst. II 52, 349–353 (2005) |

[6] | Cao, J., Yuan, K., Li, H.X.: Global asymptotical stability of generalized recurrent neural networks with multiple discrete delays and distributed delays. IEEE Trans. Neural Netw. 17(6), 1646–1651 (2006) |

[7] | Haykin, S.: Neural Networks. Prentice Hall, New York (1994) · Zbl 0828.68103 |

[8] | Liao, X., Mao, X.: Exponential stability and instability of stochastic neural networks. Stoch. Anal. Appl. 14, 165–185 (1996) · Zbl 0848.60058 |

[9] | Mao, X.: Exponential Stability of Stochastic Differential Equations. Marcel Dekker, New York (1994) · Zbl 0806.60044 |

[10] | Blythe, S., Mao, X., Liao, X.: Stability of stochastic delay neural networks. J. Franklin Inst. 338, 481–495 (2001) · Zbl 0991.93120 |

[11] | Wan, L., Sun, J.: Mean square exponential stability of stochastic delayed Hopfield neural networks. Phys. Lett. A 343, 306–318 (2005) · Zbl 1194.37186 |

[12] | Zhao, H., Ding, N.: Dynamic analysis of stochastic Cohen–Grossberg neural networks with time delays. Appl. Math. Comput. 183, 464–470 (2006) · Zbl 1117.34080 |

[13] | Wang, Z., Lauria, S., Fang, J., Liu, X.: Exponential stability of uncertain stochastic neural networks with mixed time-delays. Chaos Solitons Fractals 32, 62–72 (2007) · Zbl 1152.34058 |

[14] | Sun, Y., Cao, J.: Pth moment exponential stability of stochastic recurrent neural networks with time-varying delays. Nonlinear Anal.: Real World Appl. 8, 1171–1185 (2007) · Zbl 1196.60125 |

[15] | Sworder, D.D., Rogers, R.O.: An LQ-solution to a control problem associated with solar thermal central receiver. IEEE Trans. Autom. AC-28, 971–978 (1983) |

[16] | Willsky, A.S., Rogers, B.C.: Stochastic stability research for complex power systems. DOE Contract, LIDS, Mass. Inst. Technol., Cambridge, MA, Rep. ET-76-C-01-2295 (1979) |

[17] | Mao, X.: Exponential stability of stochastic delay interval systems with Markovian switching. IEEE Trans. Autom. Control 47, 1604–1612 (2002) · Zbl 1364.93685 |

[18] | Lou, X., Cui, B.: Delay-dependent stochastic stability of delayed Hopfield neural networks with Markovian jump parameters. J. Math. Anal. Appl. 328, 316–326 (2007) · Zbl 1132.34061 |

[19] | Wang, Z., Liu, Y., Yu, L., Liu, X.: Exponential stability of delayed recurrent neural networks with Markovian jumping parameters. Phys. Lett. A 356, 346–352 (2006) · Zbl 1160.37439 |

[20] | Huang, H., Ho, D.W.C., Qu, Y.: Robust stability of stochastic delayed additive neural networks with Markovian switching. Neural Netw. 20, 799–809 (2007) · Zbl 1125.68096 |

[21] | Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994) · Zbl 0816.93004 |

[22] | Gu, K.: An integral inequality in the stability problem of time-delay systems. In: Proceedings of 39th IEEE Conference on Decision and Control, pp. 2805–2810. Sydney, Australia (2000) |

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