×

Impulsive effects on stability of Cohen-Grossberg neural networks with variable delays. (English) Zbl 1103.34067

The authors analyze how the introduction of impulses into the continuous Cohen-Grossberg neural network with delays may affect its stability. Two questions arise: (i) Robustness of stability: whether the impulsive system remains stable when the original system is stable. (ii) Impulsive stabilization: whether the impulses are able to stabilize the original system when it is unstable. By means of two impulsive differential inequalities, sufficient conditions for the problems mentioned are given which improve and extend many previous stability criteria. These theoretical results are illustrated with a numerical study of two simple examples.

MSC:

34K20 Stability theory of functional-differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
34K45 Functional-differential equations with impulses
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Cohen, M. A.; Grossberg, S., Absolute stability of global pattern formulation and parallel memory storage by competitive neural net networks, IEEE Trans. Syst. Man Cybernet., SMC-13, 815-826 (1983) · Zbl 0553.92009
[2] Ye, H.; Michel, A. N.; Wang, K., Qualitative analysis of Cohen-Grossberg neural networks with multiple delay, Phys. Rev. E, 3, 2611-2618 (1995)
[3] Wang, L.; Zou, X. F., Exponential stability of Cohen-Grossberg neural networks, Neural Networks, 15, 415-422 (2002)
[4] Wang, L.; Zou, X. F., Harmless delays in Cohen-Grossberg neural networks, Physica D, 170, 162-173 (2002) · Zbl 1025.92002
[5] Chen, T. P.; Rong, L. B., Delay-independent stability analysis of Cohen-Grossberg neural networks, Phys. Lett. A, 317, 436-449 (2003) · Zbl 1030.92002
[6] Hwang, C. C.; Cheng, C. J.; Liao, T. L., Globally exponential stability of generalized Cohen-Grossberg neural networks with delays, Phys. Lett. A, 319, 157-166 (2003) · Zbl 1073.82597
[7] Liao, X.; Li, C.; Wong, K., Criteria for exponential stability of Cohen-Grossberg neural networks, Neural Networks, 17, 1401-1414 (2004) · Zbl 1073.68073
[8] Cao, J.; Liang, J., Boundedness and stability for Cohen-Grossberg neural network with time-varying delays, J. Math. Anal. Appl., 296, 665-685 (2004) · Zbl 1044.92001
[9] Wang, L., Stability of Cohen-Grossberg neural networks with distributed delays, Appl. Math. Comput., 160, 93-110 (2005) · Zbl 1069.34113
[10] Guan, Z. H.; Chen, G., On delayed impulsive Hopfield neural networks, Neural Networks, 12, 273-280 (1999)
[11] Guan, Z. H.; James, L.; Chen, G., On impulsive auto-associative neural networks, Neural Networks, 13, 63-69 (2000)
[12] Akca, H.; Alassar, R.; Covachev, V.; Covacheva, Z.; Al-Zahrani, E., Continuous-time additive Hopfield-type neural networks with impulses, J. Math. Anal. Appl., 290, 436-451 (2004) · Zbl 1057.68083
[13] Gopalsamy, K., Stability of artificial neural networks with impulses, Appl. Math. Comput., 154, 783-813 (2004) · Zbl 1058.34008
[14] Li, Y.; Lu, L., Global exponential stability and existence of periodic solution of Hopfield-type neural networks with impulses, Phys. Lett. A, 333, 62-71 (2004) · Zbl 1123.34303
[15] Li, Y., Global exponential stability of BAM neural networks with delays and impulses, Chaos, Solitions & Fractals, 24, 279-285 (2005) · Zbl 1099.68085
[16] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002
[17] Bainov, D. D.; Simenov, P. S., Systems with Impulse Effect: Stability Theory and Applications (1989), Ellis Horwood Limited: Ellis Horwood Limited Chichester
[18] Yang, T., Impulsive Systems and Control: Theory and Application (2001), Nova Science Publishers: Nova Science Publishers New York
[19] Ballinger, G.; Liu, X., Existence and uniqueness results for impulsive delay differential equations, Dyn. Continuous Discrete Impulsive Syst., 5, 579-591 (1999) · Zbl 0955.34068
[20] Hopfield, J., Neurons with graded response have collective computational properties like those of two-stage neurons, Proc. Natl. Acad. Sci. USA, 81, 3088-3092 (1984) · Zbl 1371.92015
[21] Driessche, P. V.D.; Zou, X. F., Global attractivity in delayed Hopfield neural networks models, SIAM J. Appl. Math., 58, 1878-1890 (1998) · Zbl 0917.34036
[22] Mohamad, S., Global exponential stability of continuous-time and discrete-time delayed bidirectional neural networks, Phys D, 159, 233-251 (2001) · Zbl 0984.92502
[23] Xu, D. Y., Asymptotic behavior of Hopfield neural networks with deays, Diff. Equat. Dyn. Syst., 9, 353-364 (2001) · Zbl 1231.34133
[24] Xu, D. Y.; Zhao, H. Y.; Zhu, H., Global dynamics of Hopfield neural networks involving variable delays, Comput. Math. Appl., 42, 39-45 (2001) · Zbl 0990.34036
[25] Wang, L. S.; Xu, D. Y., Stability for Hopfield neural with time delay, J. Vib. Control, 8, 13-18 (2002) · Zbl 1012.93054
[26] Arbib, M. A., Branins, Machines, and Mathematics (1987), Springer-Verlag: Springer-Verlag New York
[27] Haykin, S., Neural Networks: A Comprehensive Foundation (1998), Prentice-Hall: Prentice-Hall Englewood Cliffs · Zbl 0828.68103
[28] Halanay, A., Differential Equations (1966), Academic Press: Academic Press New York
[29] Gopalsamy, K., Stability and Oscillations in Delay Differential Equations of Population Dynamics (1992), Kluwer Academic Publishers: Kluwer Academic Publishers London · Zbl 0752.34039
[30] Lakshmikantham, V.; Leela, S., Differential and Integral Inequalities (1969), Academic Press: Academic Press New York · Zbl 0177.12403
[31] Berman, A.; Plemmons, R. J., Nonnegative Matrices in Mathematical Sciences (1979), Academic Press: Academic Press New York · Zbl 0484.15016
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.