Stability criteria for BAM neural networks with leakage delays and probabilistic time-varying delays.

*(English)*Zbl 1294.34075Summary: This paper is concerned with stability criteria for bidirectional associative memory (BAM) neural networks with leakage time delay and probabilistic time-varying delays. By establishing a stochastic variable with Bernoulli distribution, the information of probabilistic time-varying delay is transformed into the deterministic time-varying delay with stochastic parameters. Based on the Lyapunov-Krasovskii functional and stochastic analysis approach, delay-probability-distribution-dependent sufficient conditions are derived to achieve the globally asymptotically mean square stability of the considered BAM neural networks. The criteria are formulated in terms of a set of linear matrix inequalities (LMIs), which can be checked efficiently by use of some standard numerical packages. Finally, a numerical example and its simulations are given to demonstrate the usefulness and effectiveness of the proposed results.

##### MSC:

34K50 | Stochastic functional-differential equations |

34K20 | Stability theory of functional-differential equations |

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

PDF
BibTeX
XML
Cite

\textit{S. Lakshmanan} et al., Appl. Math. Comput. 219, No. 17, 9408--9423 (2013; Zbl 1294.34075)

Full Text:
DOI

##### References:

[1] | Haykin, S., Neural networks: A comprehensive foundation, (1998), Prentice Hall NJ · Zbl 0934.68076 |

[2] | Cichocki, A.; Unbehauen, R., Neural networks for optimization and signal processing, (1993), Wiley Chichester · Zbl 0824.68101 |

[3] | Hopfield, J. J., Neurons with graded response have collective computational properties like those of two-state neurons, Proceedings of the National Academy of Sciences, vol. 81, 3088-3092, (1984) · Zbl 1371.92015 |

[4] | Marcus, C. M.; Westervelt, R. M., Stability of analog neural networks with delay, Phys. Rev. A, 39, 347-359, (1989) |

[5] | Cao, J.; Xiao, M., Stability and Hopf bifurcation in a simplified BAM neural network with two time-delays, IEEE Trans. Neural Netw., 18, 416-430, (2007) |

[6] | Sakthivel, R.; Arunkumar, A.; Mathiyalagan, K.; Marshal Anthoni, S., Robust passivity analysis of fuzzy Cohen-Grossberg BAM neural networks with time-varying delays, Appl. Math. Comput., 218, 3799-3809, (2011) · Zbl 1257.34059 |

[7] | Mathiyalagan, K.; Sakthivel, R.; Marshal Anthoni, S., New robust passivity criteria for stochastic fuzzy BAM neural networks with time-varying delays, Commun. Nonlinear. Sci. Numer. Simul., 17, 1392-1407, (2012) · Zbl 1239.93112 |

[8] | Liu, C.; Li, C.; Duan, S., Stabilization of oscillating neural networks with time-delay by intermittent control, Int. J. Contr. Autom. Syst., 9, 1074-1079, (2011) |

[9] | Li, T.; Wang, T.; Song, A.; Fei, S., Exponential synchronization for arrays of coupled neural networks with time-delay couplings, Int. J. Contr. Autom. Syst., 9, 187-196, (2011) |

[10] | Yu, W., A LMI-based approach to global asymptotic stability of neural networks with time varying delays, Nonlinear Dyn., 48, 165-174, (2007) · Zbl 1176.92008 |

[11] | Wu, H.; Tao, F.; Qin, L.; Shi, R.; He, L., Robust exponential stability for interval neural networks with delays and non-Lipschitz activation functions, Nonlinear Dyn., 66, 479-487, (2011) · Zbl 1242.93113 |

[12] | Mathiyalagan, K.; Sakthivel, R.; Marshal Anthoni, S., Exponential stability result for discrete-time stochastic fuzzy uncertain neural networks, Phys. Lett. A, 376, 901-912, (2012) · Zbl 1255.92005 |

[13] | Park, J. H.; Kwon, O. M.; Lee, S. M., LMI optimization approach on stability for delayed neural networks of neutral-type, Appl. Math. Comput., 196, 236-244, (2008) · Zbl 1157.34056 |

[14] | Mathiyalagan, K.; Sakthivel, R.; Marshal Anthoni, S., New robust exponential stability results for discrete-time switched fuzzy neural networks with time delays, Comput. Math. Appl., 64, 2926-2938, (2012) · Zbl 1268.93118 |

[15] | Li, P.; Cao, J., Global stability in switched recurrent neural networks with time-varying delay via nonlinear measure, Nonlinear Dyn., 49, 295-305, (2007) · Zbl 1176.92003 |

[16] | Hu, S.; Wang, J., Global stability of a class of continuous-time recurrent neural networks, IEEE Trans. Neural Netw., 49, 1334-1347, (2002) · Zbl 1368.34072 |

[17] | Zhang, Z.; Zhang, T.; Huang, S.; Xiao, P., New global exponential stability result to a general Cohen-Grossberg neural networks with multiple delays, Nonlinear Dyn., 67, 2419-2432, (2012) · Zbl 1252.34088 |

[18] | Sakthivel, R.; Mathiyalagan, K.; Marshal Anthoni, S., Robust \(\mathcal{H}^\infty\) control for uncertain discrete-time stochastic neural networks with time-varying delays, IET Control Theory Appl., 6, 1220-1228, (2012) · Zbl 1268.93118 |

[19] | Chen, T.; Rong, L., Delay-independent stability analysis of Cohen-Grossberg neural networks, Phys. Lett. A, 317, 436-449, (2003) · Zbl 1030.92002 |

[20] | Kosko, B., Bidirectional associative memories, IEEE Trans. Syst. Man Cybern. B Cybern., 18, 49-60, (1988) |

[21] | Sheng, L.; Yang, H., Novel global robust exponential stability criterion for uncertain BAM neural networks with time-varying delays, Chaos Solitons Fractals, 40, 2102-2113, (2009) · Zbl 1198.93165 |

[22] | Wang, G.; Cao, J.; Xu, M., Stability analysis for stochastic BAM neural networks with Markovian jumping parameters, Neurocomputing, 72, 3901-3906, (2009) |

[23] | X.G. Liu, R. Martin, M. Wu, M.L. Tang, Global exponential stability of bidirectional associative memory neural networks with time delays, IEEE Trans. Neural Netw. 19 (2008) 397-407. |

[24] | Lou, X.; Cui, B., Stochastic exponential stability for Markovian jumping BAM neural networks with time-varying delays, IEEE Trans. Syst. Man Cybern. B Cybern., 37, 713-719, (2007) |

[25] | Arik, S., Global asymptotic stability analysis of bidirectional associative memory neural networks with time delays, IEEE Trans. Neural Netw., 16, 580-586, (2005) |

[26] | Liu, H.; Ou, Y.; Hu, J.; Liu, T., Delay-dependent stability analysis for continuous-time BAM neural networks with Markovian jumping parameters, Neural Netw., 23, 315-321, (2010) · Zbl 1400.34117 |

[27] | Liao, X.; Wong, K. W., Robust stability of interval bidirectional associative memory neural network with time delays, IEEE Trans. Syst. Man Cybern. B Cybern., 34, 1142-1154, (2004) |

[28] | Ho, D. W.C.; Liang, J.; Lam, J., Global exponential stability of impulsive high-order BAM neural networks with time-varying delays, Neural Netw., 19, 1581-1590, (2006) · Zbl 1178.68417 |

[29] | Liu, B.; Shi, P., Delay-range-dependent stability for fuzzy BAM neural networks with time-varying delays, Phys. Lett. A, 373, 1830-1838, (2009) · Zbl 1229.92004 |

[30] | Bao, H.; Cao, J., Delay-distribution-dependent state estimation for discrete-time stochastic neural networks with random delay, Neural Netw., 24, 19-28, (2011) · Zbl 1222.93213 |

[31] | Zhang, Y.; Yue, D.; Tian, E., Robust delay-distribution-dependent stability of discrete-time stochastic neural networks with time-varying delay, Neurocomputing, 72, 1265-1273, (2009) |

[32] | Yang, R.; Gao, H.; Lam, J.; Shi, P., New stability criteria for neural networks with distributed and probabilistic delays, Circuits Syst. Signal Process., 28, 505-522, (2009) · Zbl 1170.93027 |

[33] | Tang, Y.; Fang, J. A.; Xia, M.; Yu, D., Delay-distribution-dependent stability of stochastic discrete-time neural networks with randomly mixed time-varying delays, Neurocomputing, 72, 3830-3838, (2009) |

[34] | Yang, X.; Cao, J.; Lu, J., Synchronization of coupled neural networks with random coupling strengths and mixed probabilistic time-varying delays, Int. J. Robust Nonlinear Control, (2012) |

[35] | Gopalsamy, K., Leakage delays in BAM, J. Math. Anal. Appl., 325, 1117-1132, (2007) · Zbl 1116.34058 |

[36] | Peng, S., Global attractive periodic solutions of BAM neural networks with continuously distributed delays in the leakage terms, Nonlinear Anal. Real World Appl., 11, 2141-2151, (2010) · Zbl 1239.34081 |

[37] | Balasubramaniam, P.; Vembarasan, V., Asymptotic stability of BAM neural networks of neutral-type with impulsive effects and time delay in the leakage term, Int. J. Comput. Math., 88, 3271-3291, (2011) · Zbl 1247.34122 |

[38] | Balasubramaniam, P.; Kalpana, M.; Rakkiyappan, R., Global asymptotic stability of BAM fuzzy cellular neural networks with time delay in the leakage term, discrete and unbounded distributed delays, Math. Comput. Model., 53, 839-853, (2011) · Zbl 1217.34116 |

[39] | Li, C.; Huang, T., On the stability of nonlinear systems with leakage delay, J. Franklin Inst., 346, 366-377, (2009) · Zbl 1166.93367 |

[40] | Li, X.; Fu, X., Effect of leakage time-varying delay on stability of nonlinear differential systems, J. Franklin Inst., (2012) |

[41] | Li, X.; Fu, X.; Balasubramaniam, P.; Rakkiyappan, R., Existence, uniqueness and stability analysis of recurrent neural networks with time delay in the leakage term under impulsive perturbations, Nonlinear Anal. Real World Appl., 11, 4092-4108, (2010) · Zbl 1205.34108 |

[42] | Li, X.; Cao, J., Delay-dependent stability of neural networks of neutral type with time delay in the leakage term, Nonlinearity, 23, 1709-1726, (2010) · Zbl 1196.82102 |

[43] | Song, Q.; Cao, J., Passivity of uncertain neural networks with both leakage delay and time-varying delay, Nonlinear Dyn., 67, 1695-1707, (2012) · Zbl 1242.92005 |

[44] | Liu, Y.; Wang, Z.; Liu, X., Global exponential stability of generalized recurrent neural networks with discrete and distributed delays, Neural Netw., 19, 667-675, (2006) · Zbl 1102.68569 |

[45] | Wang, Z.; Shu, H.; Liu, Y.; Ho, D. W.C.; Liu, X., Robust stability analysis of generalized neural networks with discrete and distributed time delays, Chaos Solitons Fractals, 30, 886-896, (2006) · Zbl 1142.93401 |

[46] | Park, P. G.; Ko, J. W.; Jeong, C., Reciprocally convex approach to stability of systems with time-varying delays, Automatica, 47, 235-238, (2011) · Zbl 1209.93076 |

[47] | Gu, K.; Kharitonov, V.; Chen, J., Stability of time-delay systems, (2003), Birkhauser Massachusetts · Zbl 1039.34067 |

[48] | Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Publishers Dordrecht · Zbl 0752.34039 |

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.