×

zbMATH — the first resource for mathematics

Global exponential stability of generalized recurrent neural networks with discrete and distributed delays. (English) Zbl 1102.68569
Summary: This paper is concerned with the analysis problem for the global exponential stability of a class of Recurrent Neural Networks (RNNs) with mixed discrete and distributed delays. We first prove the existence and uniqueness of the equilibrium point under mild conditions, assuming neither differentiability nor strict monotonicity for the activation function. Then, by employing a new Lyapunov-Krasovskij functional, a Linear Matrix Inequality (LMI) approach is developed to establish sufficient conditions for the RNNs to be globally exponentially stable. Therefore, the global exponential stability of the delayed RNNs can be easily checked by utilizing the numerically efficient Matlab LMI toolbox, and no tuning of parameters is required. A simulation example is exploited to show the usefulness of the derived LMI-based stability conditions.

MSC:
68T05 Learning and adaptive systems in artificial intelligence
34K20 Stability theory of functional-differential equations
Software:
LMI toolbox; Matlab
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arik, S., Stability analysis of delayed neural networks, IEEE transactions on circuits systems—I, 47, 1089-1092, (2000) · Zbl 0992.93080
[2] Baldi, P.; Atiya, A.F., How delays affect neural dynamics and learning, IEEE transactions on neural networks, 5, 612-621, (1994)
[3] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in system and control theory, (1994), SIAM Philadephia · Zbl 0816.93004
[4] Cao, J., Periodic oscillation and exponential stability of delayed cnns, Physics letters A, 270, 157-163, (2000)
[5] Cao, J.; Huang, D.-S.; Qu, Y., Global robust stability of delayed recurrent neural networks, Chaos, solitons & fractals, 23, 1, 221-229, (2005) · Zbl 1075.68070
[6] Cao, J.; Wang, J., Global asymptotic stability of a general class of recurrent neural networks with time-varying delays, IEEE transactions on circuits and systems I, 50, 1, 34-44, (2003) · Zbl 1368.34084
[7] Cao, J.; Wang, J., Absolute exponential stability of recurrent neural networks with Lipschitz-continuous activation functions and time delays, Neural networks, 17, 3, 379-390, (2004) · Zbl 1074.68049
[8] Chen, B.; Wang, J., Global exponential periodicity and global exponential stability of a class of recurrent neural networks, Physics letters A, 329, 1/2, 36-48, (2004) · Zbl 1208.81063
[9] Gahinet, P.; Nemirovsky, A.; Laub, A.J.; Chilali, M., LMI control toolbox: for use with Matlab, (1995), The MATH Works, Inc. Natick, MA
[10] Gu, K. (2000). An integral inequality in the stability problem of time-delay systems. In: Proceedings of 39th IEEE conference on decision and control, Sydney, Australia, December 2000 (pp. 2805-2810).
[11] Hale, J.K., The theory of functional differential equations, (1977), Springer New York · Zbl 0425.34048
[12] Hu, S.; Wang, J., Absolute exponential stability of a class of continuous-time recurrent neural networks, IEEE transactions on neural networks, 14, 1, 35-45, (2003)
[13] Huang, H.; Cao, J., On global asymptotic stability of recurrent neural networks with time-varying delays, Applied mathematics and computation, 142, 1, 143-154, (2003) · Zbl 1035.34081
[14] Liang, J.; Cao, J., Boundedness and stability for recurrent neural networks with variable coefficients and time-varying delays, Physics letters A, 318, 1/2, 53-64, (2003) · Zbl 1037.82036
[15] Liang, J.; Cao, J., Global asymptotic stability of bi-directional associative memory networks with distributed delays, Applied mathematics and computation, 152, 415-424, (2004) · Zbl 1046.94020
[16] Liao, X.; Chen, G.; Sanchez, E.N., LMI-based approach for asymptotically stability analysis of delayed neural networks, IEEE transactions on circuits systems-I, 49, 1033-1039, (2002) · Zbl 1368.93598
[17] Liao, X.; Wong, K.W.; Li, C., Global exponential stability for a class of generalized neural networks with distributed delays, Nonlinear analysis: real world applications, 5, 527-547, (2004) · Zbl 1094.34053
[18] Nesterov, Y.; Nemirovsky, A., Interior point polynomial methods in convex programming: theory and applications, (1993), SIAM Philadelphia, PA
[19] Principle, J.C.; Kuo, J.-M.; Celebi, S., An analysis of the gamma memory in dynamic neural networks, IEEE transactions on neural networks, 5, 2, 337-361, (1994)
[20] Ruan, S.; Filfil, R.S., Dynamics of a two-neuron system with discrete and distributed delays, Physica D, 191, 323-342, (2004) · Zbl 1049.92004
[21] Tank, D.W.; Hopfield, J.J., Neural computation by concentrating information in time, Proceedings of the national Academy of sciences of the united states of America, 84, 1896-1991, (1987)
[22] Wang, Z.; Ho, D.W.C., Filtering on nonlinear time-delay stochastic systems, Automatica, 39, 1, 101-109, (2003) · Zbl 1010.93099
[23] Wang, Z.; Ho, D.W.C.; Liu, X., A note on the robust stability of uncertain stochastic fuzzy systems with time-delays, IEEE transactions on systems, man and cybernetics—part A, 34, 4, 570-576, (2004)
[24] Wang, Z.; Ho, D.W.C.; Liu, X., State estimation for delayed neural networks, IEEE transactions on neural networks, 16, 1, 279-284, (2005)
[25] Xu, S.; Lam, J.; Ho, D.W.C., Global robust exponential stability analysis for interval recurrent neural networks, Physics letters A, 325, 124-133, (2004) · Zbl 1161.93335
[26] Zeng, Z.; Wang, J.; Liao, X., Global exponential stability of a general class of recurrent neural networks with time-varying delays, IEEE transactions on circuits and systems I, 50, 10, 1353-1358, (2003) · Zbl 1368.34089
[27] Zhang, Q.; Wei, X.; Xu, J., Global exponential stability of Hopfield neural networks with continuously distributed delays, Physics letters A, 315, 6, 431-436, (2003) · Zbl 1038.92002
[28] Zhao, H., Global asymptotic stability of Hopfield neural network involving distributed delays, Neural networks, 17, 47-53, (2004) · Zbl 1082.68100
[29] Zhao, H., Existence and global attractivity of almost periodic solution for cellular neural network with distributed delays, Applied mathematics and computation, 154, 683-695, (2004) · Zbl 1057.34099
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.