×

On global exponential stability of discrete-time Hopfield neural networks with variable delays. (English) Zbl 1183.39013

Summary: Global exponential stability of a class of discrete-time Hopfield neural networks with variable delays is considered. By making use of a difference inequality, a new global exponential stability result is provided. The result only requires the delay to be bounded. For this reason, the result is milder than those presented in the earlier references. Furthermore, two examples are given to show the efficiency of our result.

MSC:

39A30 Stability theory for difference equations
39A12 Discrete version of topics in analysis
39A10 Additive difference equations
34K20 Stability theory of functional-differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Q. Zhang, X. Wei, and J. Xu, “Global exponential stability for nonautonomous cellular neural networks with delays,” Physics Letters A, vol. 351, no. 3, pp. 153-160, 2006. · Zbl 1234.34050 · doi:10.1016/j.physleta.2005.10.090
[2] Q. Zhang, X. Wei, and J. Xu, “Global asymptotic stability analysis of neural networks with time-varying delays,” Neural Processing Letters, vol. 21, no. 1, pp. 61-71, 2005. · Zbl 02225165 · doi:10.1007/s11063-004-3426-1
[3] Q. Zhang, X. Wei, and J. Xu, “Stability analysis for cellular neural networks with variable delays,” Chaos, Solitons and Fractals, vol. 28, no. 2, pp. 331-336, 2006. · Zbl 1084.34068 · doi:10.1016/j.chaos.2005.05.026
[4] Q. Zhang, X. Wei, and J. Xu, “Global asymptotic stability of cellular neural networks with infinite delay,” Neural Network World, vol. 15, no. 6, pp. 579-589, 2005.
[5] Q. Zhang, X. Wei, and J. Xu, “On global exponential stability of delayed cellular neural networks with time-varying delays,” Applied Mathematics and Computation, vol. 162, no. 2, pp. 679-686, 2005. · Zbl 1114.34337 · doi:10.1016/j.amc.2004.01.004
[6] X. Liao, G. Chen, and E. N. Sanchez, “LMI-based approach for asymptotically stability analysis of delayed neural networks,” IEEE Transactions on Circuits and Systems. I, vol. 49, no. 7, pp. 1033-1039, 2002. · Zbl 1368.93598 · doi:10.1109/TCSI.2002.800842
[7] S. Arik, “An improved global stability result for delayed cellular neural networks,” IEEE Transactions on Circuits and Systems. I, vol. 49, no. 8, pp. 1211-1214, 2002. · Zbl 1368.34083 · doi:10.1109/TCSI.2002.801264
[8] S. Arik, “An analysis of global asymptotic stability of delayed cellular neural networks,” IEEE Transactions on Neural Networks, vol. 13, no. 5, pp. 1239-1242, 2002. · doi:10.1109/TNN.2002.1031957
[9] J. Zhang and X. Jin, “Global stability analysis in delayed Hopfiled neural network models,” Neural Networks, vol. 13, no. 7, pp. 745-753, 2000. · doi:10.1016/S0893-6080(00)00050-2
[10] L. O. Chua and L. Yang, “Cellular neural networks: theory,” IEEE Transactions on Circuits and Systems. I, vol. 35, no. 10, pp. 1257-1272, 1988. · Zbl 0663.94022 · doi:10.1109/31.7600
[11] T. Roska and L. O. Chua, “Cellular neural networks with nonlinear and delay-type template elements,” in Proceedings of IEEE International Workshop on Cellular Neural Networks and Their Applications (CNNA ’90), pp. 12-25, Budapest, Hungary, December 1990. · doi:10.1109/CNNA.1990.207503
[12] S. Mohamad and K. Gopalsamy, “Exponential stability of continuous-time and discrete-time cellular neural networks with delays,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 17-38, 2003. · Zbl 1030.34072 · doi:10.1016/S0096-3003(01)00299-5
[13] H. C. Yee, P. K. Sweby, and D. F. Griffiths, “A study of spurious asymptotic numerical solutions of nonlinear differential equations by the nonlinear dynamics approach,” in Twelfth International Conference on Numerical Methods in Fluid Dynamics (Oxford, 1990), vol. 371 of Lecture Notes in Physics, pp. 259-267, Springer, Berlin, Germany, 1990.
[14] R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, River Edge, NJ, USA, 1994. · Zbl 0810.65083
[15] J. Liang, J. Cao, and J. Lam, “Convergence of discrete-time recurrent neural networks with variable delay,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 15, no. 2, pp. 581-595, 2005. · Zbl 1098.68107 · doi:10.1142/S0218127405012235
[16] W.-H. Chen, X. Lu, and D.-Y. Liang, “Global exponential stability for discrete-time neural networks with variable delays,” Physics Letters A, vol. 358, no. 3, pp. 186-198, 2006. · Zbl 1142.93393 · doi:10.1016/j.physleta.2006.05.014
[17] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992. · Zbl 0752.34039
[18] J. Cao, “Global stability conditions for delayed CNNs,” IEEE Transactions on Circuits and Systems. I, vol. 48, no. 11, pp. 1330-1333, 2001. · Zbl 1006.34070 · doi:10.1109/81.964422
[19] E. Liz and J. B. Ferreiro, “A note on the global stability of generalized difference equations,” Applied Mathematics Letters, vol. 15, no. 6, pp. 655-659, 2002. · Zbl 1036.39013 · doi:10.1016/S0893-9659(02)00024-1
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.