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Boundedness and exponential stability for nonautonomous cellular neural networks with reaction-diffusion terms. (English) Zbl 1133.35386
Summary: Employing Lyapunov functional method, we analyze the ultimate boundedness and global exponential stability of a class of reaction-diffusion cellular neural networks with time-varying delays. Some new criteria are obtained to ensure ultimate boundedness and global exponential stability of delayed reaction-diffusion cellular neural networks (DRCNNs). Without assuming that the activation functions $f_{ijl}(\cdot )$ are bounded, the results extend and improve the earlier publications.

##### MSC:
 35K57 Reaction-diffusion equations 35B35 Stability of solutions of PDE 92B20 General theory of neural networks (mathematical biology) 35R10 Partial functional-differential equations
##### Keywords:
delay; Lyapunov functional method
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##### References:
 [1] Chua, L. O.; Yang, L.: Cellular neural networks: theory. IEEE trans circuits syst I 35, 1257-1272 (1988) · Zbl 0663.94022 [2] Chua, L. O.; Yang, L.: Cellular neural networks: applications. IEEE trans circuits syst I 35, 1273-1290 (1988) [3] Roska T, Boros T, Thiran P, Chua LO. Detecting simple motion using cellular neural networks. In: Proc 1990 IEEE int workshop on cellular neural networks and their applications, 1990. p. 127-38. [4] Chua LO, Roska T. Cellular neural networks with nonlinear and delay-type template elements. In: Proc 1990 IEEE int workshop on cellular neural networks and their applications, 1990. p. 12-25. [5] Cao, J. D.; Lin, Y. P.: Stability of a class of neural network models with delay. Appl math mech 20, 851-855 (1999) · Zbl 0936.34063 [6] Cao, J. D.; Li, J. B.: The stability in neural networks with interneuronal transmission delays. Appl math mech 19, 425-430 (1998) · Zbl 0908.92003 [7] Cao, J. D.: On stability of delayed cellular neural networks. Phys lett A 261, 303-308 (1999) · Zbl 0935.68086 [8] Gopalsamy, K.; He, X. Z.: Stability in asymmetric Hopfield nets with transmission delays. Physica D 76, 344-358 (1994) · Zbl 0815.92001 [9] Liao, X.; Liao, Y.: Stability of Hopfield-type neural networks II. Sci China ser A 40, No. 8, 813-816 (1997) · Zbl 0886.68115 [10] Cao, J. D.; Wang, J.: Global asymptotic and robust stability of recurrent neural networks with time delays. IEEE trans circuits syst I: Regular papers 52, No. 2, 417-426 (2005) [11] Cao, J. D.: On exponential stability and periodic solutions of cnns with delays. Phys lett A 267, 312-318 (2000) · Zbl 1098.82615 [12] Cao, J. D.: Periodic oscillation and exponential stability of delayed cnns. Phys lett A 270, 157-163 (2000) [13] Cao, J. D.: Global exponential stability and periodic solutions of delayed cellular neural networks. J comput syst sci 60, 38-46 (2000) · Zbl 0988.37015 [14] Lou, X. Y.; Cui, B. T.: Robust stability of uncertain impulsive Hopfield neural networks with delays. Impuls dynam syst appl, DCDIS proc 3, 114-119 (2005) [15] Zhou, D. M.; Cao, J. D.: Globally exponential stability conditions for cellular neural networks with time-varying delays. Appl math comput 131, 487-496 (2002) · Zbl 1034.34093 [16] Cao, J. D.: New results concerning exponential stability and periodic solutions of delayed cellular neural networks. Phys lett A 307, No. 2-3, 136-147 (2003) · Zbl 1006.68107 [17] Cao, J. D.: Global stability conditions for delayed CNNS. IEEE trans circuits syst I 48, 1330-1333 (2001) · Zbl 1006.34070 [18] Liang, J. L.; Cao, J. D.: Boundedness and stability for recurrent neural networks with variable coefficients and time-varying delays. Phys lett A 318, 53-64 (2003) · Zbl 1037.82036 [19] Liao, X.; Fu, Y.; Gao, J.; Zhao, X.: Stability of Hopfield neural networks with reaction-diffusion terms. Acta electron sinica 28, No. 1, 78-80 (2000) [20] Cui, B. T.; Lou, X. Y.: Global asymptotic stability of BAM neural networks with distributed delays and reaction-diffusion terms. Chaos, solitons & fractals 27, No. 5, 1347-1354 (2006) · Zbl 1084.68095 [21] Liang, J. L.; Cao, J. D.: Global exponential stability of reaction-diffusion recurrent neural networks with time-varying delays. Phys lett A 314, 434-442 (2003) · Zbl 1052.82023 [22] Song, Q. K.; Zhao, Z. J.; Li, Y. M.: Global exponential stability of BAM with distributed delays and reaction-diffusion terms. Phys lett A 335, 213-225 (2005) · Zbl 1123.68347 [23] Liao, X.; Li, J.: Stability in gilpin-ayala competition models with diffusion. Nonlinear anal TMA 28, No. 10, 1751-1758 (1997) · Zbl 0872.35054 [24] Hastings, A.: Global stability in Lotka-Volterra systems with diffusion. J math biol 6, No. 2, 163-168 (1978) · Zbl 0393.92013 [25] Rothe, F.: Convergence to the equilibrium state in the Volterra-Lotka diffusion equations. J math biol 3, 319-324 (1976) · Zbl 0355.92013 [26] Jiang, H. J.; Li, Z. M.; Teng, Z. D.: Boundedness and stability for nonautonomous cellular neural networks with delay. Phys lett A 306, 313-325 (2003) · Zbl 1006.68059 [27] Zhang, Q.; Wei, X. P.; Xu, J.: Delay-dependent exponential stability of cellular neural networks with time-varying delays. Chaos, solitons & fractals 23, No. 4, 1363-1369 (2005) · Zbl 1094.34055 [28] Zhang, Q.; Wei, X. P.; Xu, J.: Stability analysis for cellular neural networks with variable delays. Chaos, solitons & fractals 28, No. 2, 331-336 (2006) · Zbl 1084.34068 [29] Burton, T. A.: Stability and periodic solutions of ordinary and functional differential equations. (1985) · Zbl 0635.34001