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Global convergence for Cohen-Grossberg neural networks with discontinuous activation functions. (English) Zbl 1261.34058

Summary: Cohen-Grossberg neural networks with discontinuous activation functions are considered. Using \(M\)-matrices and a generalized Lyapunov-like approach, the uniqueness is proved for state solutions and corresponding output solutions and equilibrium points and corresponding output equilibrium points of the considered neural networks. Moreover, global exponential stability of equilibrium points is obtained. Furthermore, by the contraction mapping principle, uniqueness and global exponential stability of the limit cycles are proved. Finally, an example is given to illustrate the effectiveness of the obtained results.

MSC:

34K20 Stability theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations
34K21 Stationary solutions of functional-differential equations
47N20 Applications of operator theory to differential and integral equations
92B20 Neural networks for/in biological studies, artificial life and related topics
34A36 Discontinuous ordinary differential equations
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[1] M. A. Cohen and S. Grossberg, “Absolute stability of global pattern formation and parallel memory storage by competitive neural networks,” IIEEE Transactions on Systems, Man, and Cybernetics, vol. 13, no. 5, pp. 815-826, 1983. · Zbl 0553.92009 · doi:10.1109/TSMC.1983.6313075
[2] K. Gopalsamy, “Global asymptotic stability in a periodic Lotka-Volterra system,” Australian Mathematical Society B, vol. 27, no. 1, pp. 66-72, 1985. · Zbl 0588.92019 · doi:10.1017/S0334270000004768
[3] X. Liao, S. Yang, S. Chen, and Y. Fu, “Stability of general neural networks with reaction-diffusion,” Science in China F, vol. 44, pp. 389-395, 2001. · Zbl 1238.93071 · doi:10.1007/BF02714741
[4] W. Lu and T. Chen, “New conditions on global stability of Cohen-Grossberg neural networks,” Neural Computation, vol. 15, no. 5, pp. 1173-1189, 2003. · Zbl 1086.68573 · doi:10.1162/089976603765202703
[5] K. Yuan and J. Cao, “An analysis of global asymptotic stability of delayed Cohen-Grossberg neural networks via nonsmooth analysis,” IEEE Transactions on Circuits and Systems I, vol. 52, no. 9, pp. 1854-1861, 2005. · Zbl 1374.34291 · doi:10.1109/TCSI.2005.852210
[6] J. Cao and X. Li, “Stability in delayed Cohen-Grossberg neural networks: LMI optimization approach,” Physica D, vol. 212, no. 1-2, pp. 54-65, 2005. · Zbl 1097.34053 · doi:10.1016/j.physd.2005.09.005
[7] C. C. Hwang, C. J. Cheng, and T. L. Liao, “Globally exponential stability of generalized Cohen-Grossberg neural networks with delays,” Physics Letters A, vol. 319, no. 1-2, pp. 157-166, 2003. · Zbl 1073.82597 · doi:10.1016/j.physleta.2003.10.002
[8] J. Cao and J. Liang, “Boundedness and stability for Cohen-Grossberg neural network with time-varying delays,” Journal of Mathematical Analysis and Applications, vol. 296, no. 2, pp. 665-685, 2004. · Zbl 1044.92001 · doi:10.1016/j.jmaa.2004.04.039
[9] J. Zhang, Y. Suda, and H. Komine, “Global exponential stability of Cohen-Grossberg neural networks with variable delays,” Physics Letters A, vol. 338, no. 1, pp. 44-50, 2005. · Zbl 1136.34347 · doi:10.1016/j.physleta.2005.02.005
[10] T. Chen and L. Rong, “Robust global exponential stability of Cohen-Grossberg neural networks with time delays,” IEEE Transactions on Neural Networks, vol. 15, no. 1, pp. 203-205, 2004. · doi:10.1109/TNN.2003.822974
[11] M. Forti and P. Nistri, “Global convergence of neural networks with discontinuous neuron activations,” IEEE Transactions on Circuits and Systems I, vol. 50, no. 11, pp. 1421-1435, 2003. · Zbl 1368.34024 · doi:10.1109/TCSI.2003.818614
[12] H. Harrer, J. A. Nossek, and R. Stelzl, “An analog implementation of discrete-time cellular neural networks,” IEEE Transactions on Neural Networks, vol. 3, no. 3, pp. 466-476, 1992. · doi:10.1109/72.129419
[13] M. P. Kennedy and L. O. Chua, “Neural networks for nonlinear programming,” IEEE Transactions on Circuits and Systems I, vol. 35, no. 5, pp. 554-562, 1988. · doi:10.1109/31.1783
[14] L. O. Chua, C. A. Desoer, and E. S. Kuh, Linear and Nonlinear Circuits, McGraw-Hill, New York, NY, USA, 1987. · Zbl 0631.94017
[15] V. I. Utkin, Sliding Modes and Their Application in Variable Structure Systems, U.S.S.R.: MIR, Moscow, Russia, 1978. · Zbl 0398.93003
[16] J.-P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin, Germany, 1984. · Zbl 0538.34007 · doi:10.1007/978-3-642-69512-4
[17] B. E. Paden and S. S. Sastry, “A calculus for computing Filippov’s differential inclusion with application to the variable structure control of robot manipulators,” IEEE Transactions on Circuits and Systems, vol. 34, no. 1, pp. 73-82, 1987. · Zbl 0632.34005 · doi:10.1109/TCS.1987.1086038
[18] M. Forti, “M-matrices and global convergence of discontinuous neural networks,” International Journal of Circuit Theory and Applications, vol. 35, no. 2, pp. 105-130, 2007. · Zbl 1131.92004 · doi:10.1002/cta.381
[19] W. Lu and T. Chen, “Dynamical behaviors of delayed neural network systems with discontinuous activation functions,” Neural Computation, vol. 18, no. 3, pp. 683-708, 2006. · Zbl 1094.68625 · doi:10.1162/neco.2006.18.3.683
[20] W. Lu and T. Chen, “Dynamical behaviors of Cohen-Grossberg neural networks with discontinuous activation functions,” Neural Networks, vol. 18, no. 3, pp. 231-242, 2005. · Zbl 1078.68127 · doi:10.1016/j.neunet.2004.09.004
[21] A. F. Filippov, “Differential equations with discontinuous right-hand side,” Transactions of the American Mathematical Society, vol. 42, pp. 199-231, 1964. · Zbl 0148.33002
[22] M. Hirsch, “Convergent activation dynamics in continuous time networks,” Neural Networks, vol. 2, pp. 331-349, 1989.
[23] D. Hershkowitz, “Recent directions in matrix stability,” Linear Algebra and its Applications, vol. 171, pp. 161-186, 1992. · Zbl 0759.15010 · doi:10.1016/0024-3795(92)90257-B
[24] H. Wu, “Global stability analysis of a general class of discontinuous neural networks with linear growth activation functions,” Information Sciences, vol. 179, no. 19, pp. 3432-3441, 2009. · Zbl 1181.34063 · doi:10.1016/j.ins.2009.06.006
[25] G. Huang and J. Cao, “Multistability of neural networks with discontinuous activation function,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 10, pp. 2279-2289, 2008. · Zbl 1221.34131 · doi:10.1016/j.cnsns.2007.07.005
[26] H. Wu and C. Shan, “Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses,” Applied Mathematical Modelling, vol. 33, no. 6, pp. 2564-2574, 2009. · Zbl 1205.34017 · doi:10.1016/j.apm.2008.07.022
[27] C. Huang and J. Cao, “Stochastic dynamics of nonautonomous Cohen-Grossberg neural networks,” Abstract and Applied Analysis, vol. 2011, Article ID 297147, 17 pages, 2011. · Zbl 1217.93175 · doi:10.1155/2011/297147
[28] X. Yang, C. Huang, D. Zhang, and Y. Long, “Dynamics of Cohen-Grossberg neural networks with mixed delays and impulses,” Abstract and Applied Analysis, vol. 2008, Article ID 432341, 14 pages, 2008. · Zbl 1160.37431 · doi:10.1155/2008/432341
[29] Q. Liu and W. Zheng, “Bifurcation of a Cohen-Grossberg neural network with discrete delays,” Abstract and Applied Analysis, vol. 2012, Article ID 909385, 11 pages, 2012. · Zbl 1238.34075 · doi:10.1155/2012/909385
[30] M. J. Park, O. M. Kwon, J. H. Park, S. M. Lee, and E. J. Cha, “Synchronization criteria for coupled stochastic neural networks with time-varying delays and leakage delay,” Journal of the Franklin Institute, vol. 349, no. 5, pp. 1699-1720, 2012. · Zbl 1254.93012 · doi:10.1016/j.jfranklin.2012.02.002
[31] O. M. Kwon, S. M. Lee, J. H. Park, and E. J. Cha, “New approaches on stability criteria for neural networks with interval time-varying delays,” Applied Mathematics and Computation, vol. 218, no. 19, pp. 9953-9964, 2012. · Zbl 1253.34066 · doi:10.1016/j.amc.2012.03.082
[32] O. M. Kwon, J. H. Park, S. M. Lee, and E. J. Cha, “A new augmented Lyapunov-Krasovskii functional approach to exponential passivity for neural networks with time-varying delays,” Applied Mathematics and Computation, vol. 217, no. 24, pp. 10231-10238, 2011. · Zbl 1225.93096 · doi:10.1016/j.amc.2011.05.021
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