# Article

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Keywords:
neutral-type; neural networks; Lyapunov functional method; stability
Summary:
By using linear matrix inequality (LMI) approach and Lyapunov functional method, we obtain some new sufficient conditions ensuring global asymptotic stability and global exponential stability of a generalized neutral-type impulsive neural networks with delays. A simulation example is provided to demonstrate the usefulness of the main results obtained. The main contribution in this paper is that a new neutral-type impulsive neural networks with variable delays is studied by constructing a novel Lyapunov functional and LMI approach.
References:
[1] Berman, A., Plemmons, R. J.: Nonnegative Matrices in Mathematical Sciences. Academic Press, New York 1979. MR 0544666
[2] Bouzerdoum, A., Pattison, T. R.: Neural networks for quadratic optimization with bound constraints. IEEE Trans. Neural Networks 4 (1993), 293-303. DOI 10.1109/72.207617
[3] Civalleri, P. P., Gilli, M., Pandolfi, L.: On stability of cellular neural networks with delay. IEEE Trans. Circuits Syst. I 40 (1993), 157-165. DOI 10.1109/81.222796 | MR 1232558
[4] Cheng, C., Liao, T., Yan, J., Hwang, C.: Globally asymptotic stability of a class of neutral-type neural networks with delays. IEEE Trans. Syst. Man Cybern. 36 (2006), 1191-1195. DOI 10.1109/tsmcb.2006.874677
[5] Cheng, L., Hou, Z., Tan, M.: A neutral-type delayed projection neural network for solving nonlinear variational inequalities. IEEE Trans. Circuits Syst. II-Express Brief 55 (2008), 806-810. DOI 10.1109/tcsii.2008.922472
[6] Chua, L. O., Yang, L.: Cellular neural networks: applications. IEEE Trans. Circuits Syst. 35 (1988), 1273-1290. DOI 10.1109/31.7601 | MR 0960778
[7] Gui, Z., Ge, W., Yang, X.: Periodic oscillation for a Hopfield neural networks with neutral delays. Phys. Lett. A 364 (2007), 267-273. DOI 10.1016/j.physleta.2006.12.013
[8] Guan, Z., Chen, G., Qin, Y.: On equilibria, stability and instability of Hopfield neural networks. IEEE Trans. Neural Networks 2 (2000), 534-540. DOI 10.1109/72.839023
[9] Hale, J.: Theory of Functional Differential Equations. Applied Mathematical Springer-Verlag, New York 1977. DOI 10.1007/978-1-4612-9892-2 | MR 0508721 | Zbl 0662.34064
[10] Hang, X.-M., Han, Q.-L.: Event-based H$_\infty$ filtering for sampled-data systems. Automatica 51 (2015), 55-69. DOI 10.1016/j.automatica.2014.10.092 | MR 3284753
[11] He, W., Chen, G., Han, Q.-L., Qian, F.: Network-based leader-following consensus of nonlinear multi-agent systems via distributed impulsive control. Inform. Sci. 20 (2017), 145-158. DOI 10.1016/j.ins.2015.06.005
[12] He, W., Qian, F., Cao, J.: Pinning-controlled synchronization of delayed neural networks with distributed-delay coupling via impulsive control. Neural Networks 85 (2017), 1-9. DOI 10.1016/j.neunet.2016.09.002
[13] He, W., Qian, F., Lam, J., Chen, G., Han, Q.-L., Kurths, J.: Quasi-synchronization of heterogeneous dynamic networks via distributed impulsive control: Error estimation, optimization and design. Automatica 62 (2015), 249-262. DOI 10.1016/j.automatica.2015.09.028 | MR 3423996
[14] Ji, D. H., 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 (2011), 6168-6175. DOI 10.1016/j.amc.2010.12.100 | MR 2773359
[15] Kaul, S. K., Liu, X. Z.: Vector Lyapunov functions for impulsive differential systems with variable times. Dyn. Continuous Discrete Impulsive Systems 6 (1999), 25-38. MR 1679754
[16] Kennedy, M. P., Chua, L. O.: Neural networks for non-linear programming. IEEE Trans. Circuits. Syst. 35 (1988), 554-562. DOI 10.1109/31.1783 | MR 0936291
[17] Lakshmikantham, V., Bainov, D. D., Simeonov, P. S.: Theory of Impulsive Differential Equations. World Scientific, Singapore 1989. DOI 10.1142/0906 | MR 1082551 | Zbl 0719.34002
[18] Lakshmikantham, V., Leela, S., Kaul, S. K.: Comparison principle for impulsive differential equations with variable times and stability theory. Nonlinear Anal. 22 (1994), 499-503. DOI 10.1016/0362-546x(94)90170-8 | MR 1266374
[19] Lakshmikantham, V., Papageorgiou, N. S., Vasundhara, J.: The method of upper and lower solutions and monotone technique for impulsive differential equations with variable moments. Appl. Anal. 15 (1993), 41-58. DOI 10.1080/00036819308840203 | MR 1278992
[20] Lakshmanan, S., Park, J. H., Jung, H. Y., Balasubramaniam, P.: Design of state estimator for neural networks with leakage, discrete and distributed delays. Appl. Math. Comput. 218 (2012), 11297-11310. DOI 10.1016/j.amc.2012.05.022 | MR 2942411
[21] Li, T., Zheng, W., Lin, C.: Delay-slope dependent stability results of recurrent neural networks. IEEE Trans. Neural Networks 22 (2011), 2138-2143. DOI 10.1109/tnn.2011.2169425
[22] Lien, C., Yu, K., Lin, Y., Chung, Y., Chung, L.: Exponential convergence rate estimation for uncertain delayed neural networks of neutral type. Chao. Solit. Fract. 40 (2009), 2491-2499. DOI 10.1016/j.chaos.2007.10.043 | MR 2533195
[23] Liu, X., Ballinger, G.: Existence and continuability of solutions for differential equations with delays and state-dependent impulses. Nonlinear Anal. 51 (2002), 633-647. DOI 10.1016/s0362-546x(01)00847-1 | MR 1920341
[24] Liu, Y., Wang, Z., Liu, X.: Global exponential stability of generalized recurrent neural networks with discrete and distributed delays. Neural Networks 19 (2006), 667-675. DOI 10.1016/j.neunet.2005.03.015
[25] Long, S., Xu, D.: Delay-dependent stability analysis for impulsive neural networks with time varying delays. Neurocomputing 71 (2008), 1705-1713. DOI 10.1016/j.neucom.2007.03.010 | MR 2466714
[26] Marcus, C. M., Westervelt, R. M.: Stability of analog neural networks with delay. Phys. Rev. A 39 (1989), 347-359. DOI 10.1103/physreva.39.347 | MR 0978323
[27] Niu, Y., Lam, J., Wang, X.: Sliding-mode control for uncertain neutral delay systems. IEE Proc. Part D: Control Theory Appl. 151 (2004), 38-44. DOI 10.1049/ip-cta:20040009
[28] Park, J. H.: Further result on asymptotic stability criterion of cellular neural networks with time-varying discrete and distributed delays. Appl. Math. Comput. 182 (2006), 1661-1666. DOI 10.1016/j.amc.2006.06.005 | MR 2282606
[29] Park, J. H., Kwon, O., Lee, S.: LMI optimization approach on stability for delayed neural networks of neutral-type. Appl. Math. Comput. 196 (2008), 236-244. DOI 10.1016/j.amc.2007.05.047 | MR 2382607
[30] Qin, J., Cao, J.: Delay-dependent robust stability of neutral-type neural networks with time delays. J. Math. Cont. Sci. Appl. 1 (2007), 179-188.
[31] Rakkiyappan, R., Balasubramaniama, P., Cao, J.: Global exponential stability results for neutral-type impulsive neural networks. Nonlinear Anal. RWA 11 (2010), 122-130. DOI 10.1016/j.nonrwa.2008.10.050 | MR 2570531
[32] Roska, T., Chua, L. O.: Cellular neural networks with nonlinear and delay-type templates. Int. J. Circuit Theory Appl. 20 (1992), 469-481. DOI 10.1002/cta.4490200504
[33] Samoilenko, A. M., Perestyuk, N. A.: Impulsive Differential Equations. World Scientific, Singapore 1995. DOI 10.1142/9789812798664
[34] Singh, V.: On global robust stability of interval Hopfield neural networks with delay. Chao. Solit. Fract. 33 (2007), 1183-1188. DOI 10.1016/j.chaos.2006.01.121 | MR 2318906
[35] Travis, C. C., Webb, G. F.: Existence and stability for partial functional differential equations. Trans. Amer. Math. Soc. 200 (1974), 395-418. DOI 10.1090/s0002-9947-1974-0382808-3 | MR 0382808
[36] Wang, Z., Wang, Y., Liu, Y.: Global synchronization for discrete-time stochastic complex networks with randomly occurred nonlinearities and mixed time delays. IEEE Trans. Neural Networks 21 (2010), 11-25. DOI 10.1109/tnn.2009.2033599
[37] Wang, J., Zhang, X.-M., Han, Q.-L.: Event-triggered generalized dissipativity filtering for neural networks with time-varying delays. IEEE Trans. Neural Networks and Learning Systems 27 (2016), 77-88. DOI 10.1109/tnnls.2015.2411734 | MR 3465626
[38] Webb, G. F.: Autonomos nonlinear functional differential equations and nonlinear semigroups. J. Math. Anal. Appl. 46 (1974), 1-12. DOI 10.1016/0022-247x(74)90277-7 | MR 0348224
[39] Xu, S., Lam, J., Ho, D., Zou, Y.: Delay-dependent exponential stability for class of neural networks with time delays. J. Comput. Appl. Math. 183 (2005), 16-28. DOI 10.1016/j.cam.2004.12.025 | MR 2156097
[40] Xu, D., Yang, Z.: Impulsive delay differential inequality and stability of neural networks. J. Math. Anal. Appl. 305 (2005), 107-120. DOI 10.1016/j.jmaa.2004.10.040 | MR 2128115
[41] Yang, Y., Cao, J.: Stability and periodicity in delayed cellular neural networks with impulsive effects. Nonlinear Anal. RWA 8 (2007), 362-374. DOI 10.1016/j.nonrwa.2005.11.004 | MR 2268091
[42] Zhang, J.: Global stability analysis in delayed cellular neural networks. Comput. Math. Appl. 45 (2003), 1707-1720. DOI 10.1016/s0898-1221(03)00149-4 | MR 1993240
[43] Zhang, X.-M., Han, Q.-L.: New Lyapunov-Krasovskii functionals for global asymptotic stability of delayed neural networks. IEEE Trans. Neural Networks 20 (2009), 533-539. DOI 10.1109/tnn.2009.2014160
[44] Zhang, X.-M., Han, Q.-L.: Global asymptotic stability for a class of generalized neural networks with interval time-varying delays. IEEE Trans. Neural Networks 22 (2011), 1180-1192. DOI 10.1109/tnn.2011.2147331 | MR 3465626
[45] Zhang, X.-M., Han, Q.-L.: Global asymptotic stability analysis for delayed neural networks using a matrix-based quadratic convex approach. Neural Networks 54 (2014), 57-69. DOI 10.1016/j.neunet.2014.02.012
[46] Zhang, X.-M., Han, Q.-L.: Event-triggered H$_\infty$ control for a class of nonlinear networked control systems using novel integral inequalities. Int. J. Robust Nonlinear Control 27 (2016), 4, 679-700. DOI 10.1002/rnc.3598 | MR 3604527
[47] Zhang, Y., Sun, J. T.: Boundedness of the solutions of impulsive differential systems with time-varying delay. Appl. Math. Comput. 154 (2004), 279-288. DOI 10.1016/s0096-3003(03)00712-4 | MR 2066196
[48] Zhang, Y., Xu, S., Chu, Y., Lu, J.: Robust global synchronization of complex networks with neutral-type delayed nodes. Appl. Math. Comput. 216 (2010), 768-778. DOI 10.1016/j.amc.2010.01.075 | MR 2606984

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