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Wu, Huaiqin; Ding, Sanbo; Guo, Xueqing; Wang, Lingling; Zhang, Luying

Robust almost periodic dynamics for interval neural networks with mixed time-varying delays and discontinuous activation functions. (English) Zbl 1432.34089

Abstr. Appl. Anal. 2013, Article ID 630623, 13 p. (2013).
Summary: The robust almost periodic dynamical behavior is investigated for interval neural networks with mixed time-varying delays and discontinuous activation functions. Firstly, based on the definition of the solution in the sense of Filippov for differential equations with discontinuous right-hand sides and the differential inclusions theory, the existence and asymptotically almost periodicity of the solution of interval network system are proved. Secondly, by constructing appropriate generalized Lyapunov functional and employing linear matrix inequality (LMI) techniques, a delay-dependent criterion is achieved to guarantee the existence, uniqueness, and global robust exponential stability of almost periodic solution in terms of LMIs. Moreover, as special cases, the obtained results can be used to check the global robust exponential stability of a unique periodic solution/equilibrium for discontinuous interval neural networks with mixed time-varying delays and periodic/constant external inputs. Finally, an illustrative example is given to demonstrate the validity of the theoretical results.

MSC:

34K14 Almost and pseudo-almost periodic solutions to functional-differential equations
93D20 Asymptotic stability in control theory

Keywords:

existence; asymptotic almost periodicity; solution of interval network system; uniqueness; global robust exponential stability; almost periodic solutions
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\textit{H. Wu} et al., Abstr. Appl. Anal. 2013, Article ID 630623, 13 p. (2013; Zbl 1432.34089)
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References:

[1] Forti, M.; Nistri, P., Global convergence of neural networks with discontinuous neuron activations, IEEE Transactions on Circuits and Systems I, 50, 11, 1421-1435 (2003) · Zbl 1368.34024
[2] Forti, M.; Grazzini, M.; Nistri, P.; Pancioni, L., Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations, Physica D, 214, 1, 88-99 (2006) · Zbl 1103.34044
[3] Forti, M., M-matrices and global convergence of discontinuous neural networks, International Journal of Circuit Theory and Applications, 35, 2, 105-130 (2007) · Zbl 1131.92004
[4] Lu, W.; Chen, T., Dynamical behaviors of Cohen-Grossberg neural networks with discontinuous activation functions, Neural Networks, 18, 3, 231-242 (2005) · Zbl 1078.68127
[5] Forti, M.; Nistri, P.; Papini, D., Global exponential stability and global convergence in finite time of delayed neural networks with infinite gain, IEEE Transactions on Neural Networks, 16, 6, 1449-1463 (2005)
[6] Li, L.; Huang, L., Global asymptotic stability of delayed neural networks with discontinuous neuron activations, Neurocomputing, 72, 16-18, 3726-3733 (2009)
[7] Qin, S.; Xue, X., Global exponential stability and global convergence in finite time of neural networks with discontinuous activations, Neural Processing Letters, 29, 3, 189-204 (2009)
[8] Meng, Y.; Huang, L.; Guo, Z.; Hu, Q., Stability analysis of Cohen-Grossberg neural networks with discontinuous neuron activations, Applied Mathematical Modelling, 34, 2, 358-365 (2010) · Zbl 1185.34106
[9] Li, L.; Huang, L., Dynamical behaviors of a class of recurrent neural networks with discontinuous neuron activations, Applied Mathematical Modelling, 33, 12, 4326-4336 (2009) · Zbl 1178.37126
[10] Nie, X.; Cao, J., Existence and global stability of equilibrium point for delayed competitive neural networks with discontinuous activation functions, International Journal of Systems Science, 43, 3, 459-474 (2012) · Zbl 1259.93104
[11] Liu, J.; Liu, X.; Xie, W.-C., Global convergence of neural networks with mixed time-varying delays and discontinuous neuron activations, Information Sciences, 183, 92-105 (2012) · Zbl 1245.93118
[12] Liu, X.; Chen, T.; Cao, J.; Lu, W., Dissipativity and quasi-synchronization for neural networks with discontinuous activations and parameter mismatches, Neural Networks, 24, 1013-1021 (2011) · Zbl 1264.93048
[13] Liu, X.; Cao, J.; Yu, W., Filippov systems and quasi-synchronization control for switched networks, Chaos, 22 (2012) · Zbl 1319.93034
[14] Bao, G.; Zeng, Z., Analysis and design of associative memories based on recurrent neural network with discontinuous activation functions, Neurocomputing, 77, 101-107 (2012)
[15] Chen, X.; Huang, H.; Guo, Z., Finite time stability of periodic solution for hopfield neural networks with discontinuous activations, Neurocomputing, 103, 43-49 (2013)
[16] Cai, Z.; Huang, L.; Guo, Z.; Chen, X., On the periodic dynamics of a class of time-varying delayed neural networks via differential inclusions, Neural Networks, 33, 97-113 (2012) · Zbl 1269.34076
[17] Cai, Z.; Huang, L., Existence an global asymptotic stability of periodic solution for discrete and distributed time-varying delayed neural networks with discontinuous activations, Neurocomputing, 74, 3170-3179 (2011)
[18] Liu, X.; Cao, J., On periodic solutions of neural networks via differential inclusions, Neural Networks, 22, 4, 329-334 (2009) · Zbl 1335.93058
[19] Huang, L.; Wang, J.; Zhou, X., Existence and global asymptotic stability of periodic solutions for Hopfield neural networks with discontinuous activations, Nonlinear Analysis: Real World Applications, 10, 3, 1651-1661 (2009) · Zbl 1160.92002
[20] Liu, X.; Cao, J.; Huang, G., Complete periodic synchronization of delayed neural networks with discontinuous activations, International Journal of Bifurcation and Chaos, 20, 7, 2151-2164 (2010) · Zbl 1196.34063
[21] Wu, H.; Li, Y., Existence and stability of periodic solution for BAM neural networks with discontinuous neuron activations, Computers & Mathematics with Applications, 56, 8, 1981-1993 (2008) · Zbl 1165.34351
[22] Wu, H., Stability analysis for periodic solution of neural networks with discontinuous neuron activations, Nonlinear Analysis: Real World Applications, 10, 3, 1717-1729 (2009) · Zbl 1160.92004
[23] Wu, H., Global stability analysis of a general class of discontinuous neural networks with linear growth activation functions, Information Sciences, 179, 19, 3432-3441 (2009) · Zbl 1181.34063
[24] Cheng, Y.; Cong, F.; Hua, H., Anti-periodic solutions for nonlinear evolution equations, Advances in Difference Equations, 2012, article 165 (2012) · Zbl 1380.34061
[25] Wu, H.; Xue, X.; Zhong, X., Stability analysis for neural networks with discontinuous neuron activations and impulses, International Journal of Innovative Computing, Information and Control, 3, 6B, 1537-1548 (2007)
[26] Wu, H.; Shan, C., Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses, Applied Mathematical Modelling, 33, 6, 2564-2574 (2009) · Zbl 1205.34017
[27] Papini, D.; Taddei, V., Global exponential stability of the periodic solution of a delayed neural network with discontinuous activations, Physics Letters A, 343, 1-3, 117-128 (2005) · Zbl 1194.34131
[28] Chen, X.; Song, Q., Global exponential stability of the periodic solution of delayed Cohen-Grossberg neural networks with discontinuous activations, Neurocomputing, 73, 16-18, 3097-3104 (2010) · Zbl 1197.34084
[29] He, X.; Lu, W.; Chen, T., Nonnegative periodic dynamics of delayed Cohen-Grossberg neural networks with discontinuous activations, Neurocomputing, 73, 13-15, 2765-2772 (2010)
[30] Allegretto, W.; Papini, D.; Forti, M., Common asymptotic behavior of solutions and almost periodicity for discontinuous, delayed, and impulsive neural networks, IEEE Transactions on Neural Networks, 21, 7, 1110-1125 (2010)
[31] Lu, W.; Chen, T., Almost periodic dynamics of a class of delayed neural networks with discontinuous activations, Neural Computation, 20, 4, 1065-1090 (2008) · Zbl 1146.68422
[32] Qin, S.; Xue, X.; Wang, P., Global exponential stability of almost periodic solution of delayed neural networks with discontinuous activations, Information Sciences, 220, 367-378 (2013) · Zbl 1291.92015
[33] Wang, J.; Huang, L., Almost periodicity for a class of delayed Cohen-Grossberg neural networks with discontinuous activations, Chaos Solitons Fractals, 45, 1157-1170 (2012) · Zbl 1258.92002
[34] Zuo, Y.; Wang, Y.; Huang, L.; Wang, Z.; Liu, X.; Wu, X., Robust stability criterion for delayed neural networks with discontinuous activation functions, Neural Processing Letters, 29, 1, 29-44 (2009)
[35] Wu, X.; Wang, Y.; Huang, L.; Zuo, Y., Robust exponential stability criterion for uncertain neural networks with discontinuous activation functions and time-varying delays, Neurocomputing, 73, 7-9, 1265-1271 (2010)
[36] Guo, Z.; Huang, L., LMI conditions for global robust stability of delayed neural networks with discontinuous neuron activations, Applied Mathematics and Computation, 215, 3, 889-900 (2009) · Zbl 1187.34098
[37] Liu, X.; Cao, J., Robust state estimation for neural networks with discontinuous activations., IEEE Transactions on Systems, Man, and Cybernetics. Part B, 40, 6, 1425-1437 (2010)
[38] Clarke, F. H., Oprimization and Non-Smooth Analysis (1983), New York, NY, USA: Woley, New York, NY, USA
[39] Filippov, A. F., Differential Equations with Discontinuous Right-Hand Side, Mathematics and Its Applications. Differential Equations with Discontinuous Right-Hand Side, Mathematics and Its Applications, Mathematics and Its Applications (Soviet Series) (1984), Boston, Mass, USA: Kluwer Academic, Boston, Mass, USA · Zbl 0138.32204
[40] Aubin, J.-P.; Cellina, A., Differential Inclusions. Differential Inclusions, Grundlehren der Mathematischen Wissenschaften, 264, xiii+342 (1984), Berlin, Germany: Springer, Berlin, Germany
[41] Levitan, B. M.; Zhikov, V. V., Almost Periodic Functions and Differential Equations, xi+211 (1982), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0499.43005
[42] Yang, X.; Huang, C.; Zhu, Q., Synchronization of switched neural networks with mixed delays via impulsive control, Chaos, Solitons & Fractals, 44, 10, 817-826 (2011) · Zbl 1268.93013
[43] Ge, X., Applied Functional Analysis (1996), Hangzhou, China: Zhejiang University, Hangzhou, China
[44] Wang, Z.; Wang, Y.; Liu, Y., Global synchronization for discrete-time stochastic complex networks with randomly occurred nonlinearities and mixed time delays, IEEE Transactions on Neural Networks, 21, 1, 11-25 (2010)
[45] Hu, J.; Wang, Z.; Niu, Y.; Stergioulas, L. K., \(H_\infty\) sliding mode observer design for a class of nonlinear discrete time-delay systems: a delay-fractioning approach, International Journal of Robust and Nonlinear Control, 22, 16, 1806-1826 (2012) · Zbl 1273.93032
[46] Hu, J.; Wang, Z.; Gao, H.; Stergioulas, L., Robust sliding mode control for discrete stochastic systems with mixed time delays, randomly occurring uncertainties, and randomly occurring nonlinearities, IEEE Transactions on Industrial Electronics, 59, 3008-3015 (2012)
[47] Hu, J.; Wang, Z.; Gao, H., A delay fractioning approach to robust sliding mode control for discrete-time stochastic systems with randomly occurring non-linearities, IMA Journal of Mathematical Control and Information, 28, 3, 345-363 (2011) · Zbl 1228.93025
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