# zbMATH — the first resource for mathematics

Discontinuous Lyapunov functional approach to synchronization of time-delay neural networks using sampled-data. (English) Zbl 1263.34075
Summary: This paper investigates the synchronization problem of neural networks with time-varying delay under sampled-data control in the presence of a constant input delay. Based on the extended Wirtinger inequality, a discontinuous Lyapunov functional is introduced, which makes full use of the sawtooth structure characteristic of sampling input delay. A simple and less conservative synchronization criterion is given to ensure the master systems synchronize with the slave systems by using the linear matrix inequality (LMI) approach. The design method of the desired sampled-data controller is also proposed. Finally, two numerical examples are given to illustrate the effectiveness of the proposed methods.

##### MSC:
 34D06 Synchronization of solutions to ordinary differential equations 34H10 Chaos control for problems involving ordinary differential equations 92B20 Neural networks for/in biological studies, artificial life and related topics
Full Text:
##### References:
 [1] Zhang, H., Ma, T., Huang, G., Wang, Z.: Robust global exponential synchronization of uncertain chaotic delayed neural networks via dual-stage impulsive control. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 40, 831–844 (2010) [2] Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990) · Zbl 0938.37019 [3] Cao, J., Li, H., Ho, D.W.C.: Synchronization criteria of Lur’e systems with time-delay feedback control. Chaos Solitons Fractals 23, 1285–1298 (2005) · Zbl 1086.93050 [4] Xiong, W., Xie, W., Cao, J.: Adaptive exponential synchronization of delayed chaotic networks. Physica A 370, 832–842 (2006) [5] Karimi, H.R.: Robust synchronization and fault detection of uncertain master-slave systems with mixed time-varying delays and nonlinear perturbations. Int. J. Control. Autom. Syst. 9, 671–680 (2011) [6] Li, T., Wang, T., Song, A., Fei, S.: Exponential synchronization for arrays of coupled neural networks with time-delay couplings. Int. J. Control. Autom. Syst. 9, 187–196 (2011) [7] Lee, S.M., Kwon, O.M., Park, J.H.: Regional asymptotic stability analysis for discrete-time delayed systems with saturation nonlinearity. Nonlinear Dyn. 67, 885–892 (2012) · Zbl 1242.93122 [8] Kwon, O.M., Park, J.H., Lee, S.M.: Secure communication based on chaotic synchronization via interval time-varying delay feedback control. Nonlinear Dyn. 63, 239–252 (2011) · Zbl 1215.93127 [9] Lee, S.M., Choi, S.J., Ji, D.H., Park, J.H., Won, S.C.: Synchronization for chaotic Lur’e systems with sector restricted nonlinearities via delayed feedback control. Nonlinear Dyn. 59, 277–288 (2010) · Zbl 1183.70073 [10] Han, Q.L.: On designing time-varying delay feedback controllers for master-slave synchronization of Lur’e systems. IEEE Trans. Circuits Syst. I, Regul. Pap. 54, 1573–1583 (2007) · Zbl 1374.93299 [11] Lee, S.M., Ji, D.H., Park, J.H., Won, S.C.: $$$\backslash$$mathcal{H}_{$$\backslash$$infty}$ synchronization of chaotic systems via dynamic feedback approach. Phys. Lett. A 372, 4905–4912 (2008) · Zbl 1221.93087 [12] Huang, H., Feng, G., Cao, J.: Exponential synchronization of chaotic Lur’e systems with delayed feedback control. Nonlinear Dyn. 57, 441–453 (2009) · Zbl 1176.70034 [13] Miao, Q., Tang, Y., Lu, S., Fang, J.: Lag synchronization of a class of chaotic systems with unknown parameters. Nonlinear Dyn. 57, 107–112 (2009) · Zbl 1176.34097 [14] Zhang, J., Tang, W.: Control and synchronization for a class of new chaotic systems via linear feedback. Nonlinear Dyn. 58, 675–686 (2009) · Zbl 1183.70075 [15] Park, J.H., Kwon, O.M.: Synchronization of cellular neural networks of neutral type via dynamic feedback controller. Chaos Solitons Fractals 42, 1299–1304 (2009) · Zbl 1198.93182 [16] Zhang, H., Huang, W., Wang, Z., Chai, T.: Adaptive synchronization between two different chaotic systems with unknown parameters. Phys. Lett. A 350, 363–366 (2006) · Zbl 1195.93121 [17] Zhang, C., He, Y., Wu, M.: Improved global asymptotical synchronization of chaotic Lur’e systems with sampled-data control. IEEE Trans. Circuits Syst. II, Express Briefs 56, 320–324 (2009) [18] Li, P., Cao, J., Wang, Z.: Robust impulsive synchronization of coupled delayed neural networks with uncertainties. Physica A 373, 261–272 (2007) [19] Lu, J., Cao, J., Ho, D.W.C.: Adaptive stabilization and synchronization for chaotic Lur’e systems with time-varying delay. IEEE Trans. Circuits Syst. I, Regul. Pap. 55, 1347–1356 (2008) [20] Balasubramaniam, P., Chandran, R., Theesar, S.J.S.: Synchronization of chaotic nonlinear continuous neural networks with time-varying delay. Cogn. Neurodyn. 5, 361–371 (2011) [21] Gupta, M.M., Jin, L., Homma, N.: Static and Dynamic Neural Networks: From Fundamentals to Advanced Theory. Wiley/IEEE Press, New York (2003) [22] Wang, Z., Zhang, H.: Global asymptotic stability of reaction–diffusion Cohen–Grossberg neural networks with continuously distributed delays. IEEE Trans. Neural Netw. 20, 39–49 (2010) [23] Kwon, O.M., Park, J.H., Lee, S.M.: Augmented Lyapunov functional approach to stability of uncertain neutral systems with time-varying delays. Appl. Math. Comput. 207, 202–212 (2009) · Zbl 1178.34091 [24] Park, J.H., Kwon, O.M.: Further results on state estimation for neural networks of neutral-type with time-varying delay. Appl. Math. Comput. 208, 69–75 (2009) · Zbl 1169.34334 [25] Park, M.J., Kwon, O.M., Park, J.H., Lee, S.M.: A new augmented Lyapunov–Krasovskii functional approach for stability of linear systems with time-varying delays. Appl. Math. Comput. 217, 7197–7209 (2011) · Zbl 1219.93106 [26] Ji, D., Koo, J.H., Won, S.C., Lee, S.M., Park, J.H.: Passivity-based control for Hopfield neural networks using convex representation. Appl. Math. Comput. 217, 6168–6175 (2011) · Zbl 1209.93056 [27] Kwon, O.M., Lee, S., Park, J.H.: Improved delay-dependent exponential stability for uncertain stochastic neural networks with time-varying delays. Phys. Lett. A 374, 1232–1241 (2010) · Zbl 1236.92006 [28] He, Y., Liu, G., Rees, D.: New delay-dependent stability criteria for neural networks with time-varying delay. IEEE Trans. Neural Netw. 18, 310–314 (2007) [29] Wu, L., Feng, Z., Zheng, W.: Exponential stability analysis for delayed neural networks with switching parameters: average dwell time approach. IEEE Trans. Neural Netw. 21, 1396–1407 (2010) [30] Liu, Y., Wang, Z., Serrano, A., Liu, X.: Discrete-time recurrent neural networks with time-varying delays: exponential stability analysis. Phys. Lett. A 362, 480–488 (2007) [31] Xu, S., Lam, J., Ho, D.W.C., Zou, Y.: Improved global robust asymptotic stability criteria for delayed cellular neural networks. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 35, 1317–1321 (2005) [32] Zhang, H., Liu, Z., Huang, G., Wang, Z.: Novel weighting-delay-based stability criteria for recurrent neural networks with time-varying delay. IEEE Trans. Neural Netw. 21, 91–106 (2010) [33] Balasubramaniam, P., Lakshmanan, S., Theesar, S.J.S.: State estimation for Markovian jumping recurrent neural networks with interval time-varying delays. Nonlinear Dyn. 60, 661–675 (2010) · Zbl 1194.62109 [34] Feng, Z., Lam, J.: Stability and dissipativity analysis of distributed delay cellular neural networks. IEEE Trans. Neural Netw. 22, 976–981 (2011) [35] Liu, X., Chen, T., Cao, J., Lu, W.: Dissipativity and quasi-synchronization for neural networks with discontinuous activations and parameter mismatches. Neural Netw. 24, 1013–1021 (2011) · Zbl 1264.93048 [36] Liu, X., Cao, J.: Local synchronization of one-to-one coupled neural networks with discontinuous activations. Cogn. Neurodyn. 5, 13–20 (2011) [37] Yu, W., Cao, J.: Synchronization control of stochastic delayed neural networks. Physica A 373, 252–260 (2007) [38] Karimi, H.R., Maass, P.: Delay-range-dependent exponential $$$\backslash$$mathcal{H}_{$$\backslash$$infty}$ synchronization of a class of delayed neural networks. Chaos Solitons Fractals 41, 1125–1135 (2009) · Zbl 1198.93179 [39] Qi, D., Liu, M., Qiu, M., Zhang, S.: Exponential $$$\backslash$$mathcal{H}_{$$\backslash$$infty}$ synchronization of general discrete-time chaotic neural networks with or without time delays. IEEE Trans. Neural Netw. 21, 1358–1365 (2010) [40] Zhang, C., He, Y., Wu, M.: Exponential synchronization of neural networks with time-varying mixed delays and sampled-data. Neurocomputing 74, 265–273 (2010) · Zbl 05849746 [41] Liu, K., Fridman, E.: Wirtinger’s inequality and Lyapunov-based sampled-data stabilization. Automatica 48, 102–108 (2012) · Zbl 1244.93094 [42] Gao, H., Chen, T., Lam, J.: A new delay system approach to network-based control. Automatica 44, 39–52 (2008) · Zbl 1138.93375 [43] Wang, Y., Zhang, H., Wang, X., Yang, D.: Networked synchronization control of coupled dynamic networks with time-varying delay. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 40, 1468–1479 (2010) [44] Yue, D., Han, Q., Lam, J.: Network-based robust $$$\backslash$$mathcal{H}_{$$\backslash$$infty}$ control of systems with uncertainty. Automatica 41, 999–1007 (2005) · Zbl 1091.93007 [45] 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 [46] Gu, K., Kharitonov, V.K., Chen, J.: Stability of Time-Delay Systems. Birkhäuser, Boston (2003) · Zbl 1039.34067 [47] Liu, K., Suplin, V., Fridman, E.: Stability of linear systems with general sawtooth delay. IMA J. Math. Control Inf. 27, 419–436 (2011) · Zbl 1206.93080 [48] 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
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.