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Finite-time stability analysis of fractional-order complex-valued memristor-based neural networks with time delays. (English) Zbl 1331.34154
Summary: In this paper, the problem of finite-time stability of fractional-order complex-valued memristor-based neural networks (NNs) with time delays is extensively investigated. We first initiate the fractional-order complex-valued memristor-based NNs with the Caputo fractional derivatives. Using the theory of fractional-order differential equations with discontinuous right-hand sides, Laplace transforms, Mittag-Leffler functions and generalized Gronwall inequality, some new sufficient conditions are derived to guarantee the finite-time stability of the considered fractional-order complex-valued memristor-based NNs. In addition, some sufficient conditions are also obtained for the asymptotical stability of fractional-order complex-valued memristor-based NNs. Finally, a numerical example is presented to demonstrate the effectiveness of our theoretical results.

MSC:
34K37 Functional-differential equations with fractional derivatives
34K20 Stability theory of functional-differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
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[1] Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999) · Zbl 0924.34008
[2] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Application of Fractional Differential Equations. Elsevier, New York (2006) · Zbl 1092.45003
[3] Lundstrom, B; Higgs, M; Spain, W; Fairhall, A, Fractional differentiation by neocortical pyramidal neurons, Nat. Neurosci., 11, 1335-1342, (2008)
[4] Arena, P; Fortuna, L; Porto, D, Chaotic behavior in non-integer-order cellular neural networks, Phys. Rev. E, 61, 776-781, (2000)
[5] Anastassiou, G, Fractional neural network approximation, Comput. Math. Appl., 64, 1655-1676, (2012) · Zbl 1268.41007
[6] Kaslik, E; Sivasundaram, S, Nonlinear dynamics and chaos in fractional-order neural networks, Neural Netw., 32, 245-256, (2012) · Zbl 1254.34103
[7] Yu, J; Hu, C; Jiang, H, \(α \)-stability and \(α \)-synchronization for fractional-order neural networks, Neural Netw., 35, 82-87, (2012) · Zbl 1258.34118
[8] Chen, L., Chai, Y., Wu, R., Ma, T., Zhai, H.: Dynamic analysis of a class of fractional-order neural networks with delay. Neurocomputing 111, 190-194 (2013) · Zbl 1154.93377
[9] Yang, CG; Ge, SS; Xiang, C; Chai, T; Lee, TH, Output feedback NN control for two classes of discrete-time systems with unknown control directions in a unified approach, IEEE Trans. Neural Netw., 19, 1873-1886, (2008)
[10] Liu, YJ; Chen, CLP; Wen, GX; Tong, SC, Adaptive neural output feedback tracking control for a class of uncertain discrete-time nonlinear systems, IEEE Trans. Neural Netw., 22, 1162-1167, (2011)
[11] Yang, CG; Ge, SS; Lee, TH, Output feedback adaptive control of a class of nonlinear discrete-time systems with unknown control directions, Automatica, 45, 270-276, (2009) · Zbl 1154.93377
[12] Ge, SS; Yang, CG; Lee, TH, Adaptive predictive control using neural network for a class of pure-feedback systems in discrete time, IEEE Trans. Neural Netw., 19, 1599-1614, (2008) · Zbl 1151.35415
[13] Li, Y; Yang, CG; Ge, SS; Lee, TH, Adaptive output feedback NN control of a class of discrete-time MIMO nonlinear systems with unknown control directions, IEEE Trans. Syst. Man Cybern. B, 41, 507-517, (2011)
[14] Liu, Y.J., Tong, S.C., Wang, D., Li, T.S., Chen, C.L.P.: Adaptive neural output feedback controller design with reduced-order observer for a class of uncertain nonlinear SISO systems. IEEE Trans. Neural Netw. 22, 1328-1334 (2011)
[15] Hirose, A.: Complex-Valued Neural Networks. Springer, Berlin (2012) · Zbl 1235.68016
[16] Nitta, T, Orthogonality of decision boundaries of complex-valued neural networks, Neural Comput., 16, 73-97, (2004) · Zbl 1084.68105
[17] Tanaka, G; Aihara, K, Complex-valued multistate associative memory with nonlinear multilevel functions for gray-level image reconstruction, IEEE Trans. Neural Netw., 20, 1463-1473, (2009)
[18] Hu, J; Wang, J, Global stability of complex-valued recurrent neural networks with time-delays, IEEE Trans. Neural Netw. Learn. Syst., 23, 853-865, (2012)
[19] Duan, C; Song, Q, Boundedness and stability for discrete-time delayed neural network with complex-valued linear threshold neurons, Discrete Dyn. Nat. Soc., 2010, 368379, (2010) · Zbl 1200.39006
[20] Liu, X., Fang, K., Liu, B.: A synthesis method based on stability analysis for complex-valued Hopeld neural network. In Asian Control Conference, 2009. ASCC 2009, 7th. IEEE, pp. 1245-1250 (2009)
[21] Kuroe, Y., Yoshid, M., Mori, T.: On activation functions for complex-valued neural networks-existence of energy functions. In: Artificial Neural Networks and Neural Information Processing, pp. 985-992. Springer, New York (2003) · Zbl 1037.68669
[22] Zhou, B; Song, Q, Boundedness and complete stability of complex-valued neural networks with time delay, IEEE Trans. Neural Netw. Learn. Syst., 24, 1227-1238, (2013)
[23] Bohner, M; Rao, VSH; Sanyal, S, Global stability of complex-valued neural networks on time scales, Differ. Equ. Dyn. Syst., 19, 3-11, (2011) · Zbl 1258.34180
[24] Chen, X; Song, Q, Global stability of complex-valued neural networks with both leakage time delay and discrete time delay on time scales, Neurocomputing, 121, 254-264, (2013)
[25] Xu, X; Zhang, J; Shi, J, Exponential stability of complex-valued neural networks with mixed delays, Neurocomputing, 128, 483-490, (2014)
[26] Fang, T., Sun, J.: Further investigate the stability of complex-valued recurrent neural networks with time-delays. IEEE Trans. Neural Netw. Learn. Syst. (2014). doi:10.1109/TNNLS.2013.2294638
[27] Chua, LO, Memristor-the missing circuit element, IEEE Trans. Circuit Theory, 18, 507-519, (1971)
[28] Tour, JM; He, T, The fourth element, Nature, 453, 42-43, (2008)
[29] Strukov, DB; Snider, GS; Sterwart, DR; Williams, RS, The missing memristor found, Nature, 453, 80-83, (2008)
[30] Wang, X; Chen, Y; Xi, H; Li, H; Dimitrov, D, Spintronic memristor through spin-torque-induced magnetization motion, IEEE Electron Device Lett., 30, 294-297, (2009)
[31] Itoh, M; Chua, LO, Memristor cellular automata and memristor discrete-time cellular neural networks, Int. J. Bifurc. Chaos, 19, 3605-3656, (2009) · Zbl 1182.37014
[32] Hu, J., Wang, J.: Global uniform asymptotic stability of memristor-based recurrent neural networks with time delay. In: International Joint Conference on Neural Networks (IJCNN 2010), pp. 1-8. Barcelona (2010)
[33] Wu, A; Zeng, Z, Dynamic behaviors of memristor-based recurrent neural networks with time-varying delays, Neural Netw., 36, 1-10, (2012) · Zbl 1258.34165
[34] Wu, A; Zeng, Z, Passivity analysis of memristive neural networks with different memductance functions, Commun. Nonlinear Sci. Numer. Simulat., 19, 274-285, (2014) · Zbl 1344.93091
[35] Cai, Z; Huang, L, Functional differential inclusions and dynamic behaviors for memristor-based BAM neural networks with time-varying delays, Commun. Nonlinear Sci. Numer. Simulat., 19, 1279-1300, (2014)
[36] Wang, W; Li, L; Peng, H; Xiao, J; Yang, Y, Synchronization control of memristor-based recurrent neural networks with perturbations, Neural Netw., 53, 8-14, (2014) · Zbl 1307.93038
[37] Wu, A; Zeng, Z; Fu, C, Dynamic analysis of memristive neural system with unbounded time-varying delays, J. Frankl. Inst., 351, 3032-3041, (2014) · Zbl 1372.94486
[38] Yang, X; Cao, J; Yu, W, Exponential synchronization of memristive Cohen-Grossberg neural networks with mixed delays, Cogn. Neurodyn., 8, 239-249, (2014)
[39] Chen, J; Zeng, Z; Jiang, P, Global Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks, Neural Netw., 51, 1-8, (2014) · Zbl 1313.35053
[40] Wang, G; Cao, J; Liang, J, Exponential stability in the mean square for stochastic neural networks with mixed time-delays and Markovian jumping parameters, Nonlinear Dyn., 57, 209-218, (2009) · Zbl 1176.92007
[41] Li, X; Cao, J, Delay-independent exponential stability of stochastic Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms, Nonlinear Dyn., 50, 363-371, (2007) · Zbl 1176.92004
[42] Cheng, Z; Cao, J, Bifurcation and stability analysis of a neural network model with distributed delays, Nonlinear Dyn., 46, 363-373, (2006) · Zbl 1169.92001
[43] Huang, X; Cao, J; Ho, DWC, Existence and attractivity of almost periodic solution for recurrent neural networks with unbounded delays and variable coefficients, Nonlinear Dyn., 45, 337-351, (2006) · Zbl 1130.68084
[44] Cao, J; Wan, Y, Matrix measure strategies for stability and synchronization of inertial BAM neural network with time delays, Neural Netw., 53, 165-172, (2014) · Zbl 1322.93087
[45] Li, P; Cao, J, Global stability in switched recurrent neural network with time-varying delay via nonlinear measure, Nonlinear Dyn., 49, 295-305, (2007) · Zbl 1176.92003
[46] Lazarevic, M., Debeljkovic, D., Nenadic, Z., Milinkovic, S.: Finite time stability of time delay systems. IMA J. Math. Control Inf. 17, 101-109 (2000) · Zbl 0979.93095
[47] Debeljkovic, D., Lazarevic, M., Koruga, D., Milinkovic, S., Jovanovic, M., Jacic, L.: Further results on non-lyapunov stability of the linear nonautonomous systems with delayed state. Facta Univ. Ser. Mech. Automat. Control Robot. 3, 231-241 (2001)
[48] Chen, X; Huang, L; Guo, Z, Finite time stability of periodic solution for Hopfield neural networks with discontinuous activations, Neurocomputing, 103, 43-49, (2013)
[49] Lazarevic, MP; Spasic, AM, Finite-time stability analysis of fractional order time-delay systems: gronwalls approach, Math. Comput. Model., 49, 475-481, (2009) · Zbl 1165.34408
[50] Denghao, P; Wei, J, Finite-time stability analysis of neutral fractional time-delay systems via generalized gronwalls inequality, Abstr. Appl. Anal., 2014, 610547, (2014) · Zbl 1406.93237
[51] Wu, R., Hei, X., Chen, L.: Finite-time stability of fractional-order neural networks with delay. Commun. Theor. Phys. 60, 189-193 (2013) · Zbl 1284.92016
[52] Alofi, A., Cao, J., Elaiw, A., Al-Mozrooei, A.: Delay-dependent stability criterion of caputo fractional neural networks with distributed delay. Discrete Dyn. Nat. Soc. 2014, 529358 (2014) · Zbl 1169.92001
[53] Ye, H; Gao, J; Ding, Y, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328, 1075-1081, (2007) · Zbl 1120.26003
[54] Filippov, A.F.: Differential Equations with Discontinuous Right-Hand Sides. Mathematics and its Applications. Kluwer, Boston (1988) · Zbl 0664.34001
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