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Stability analysis of impulsive Cohen-Grossberg neural networks with distributed delays and reaction-diffusion terms. (English) Zbl 1168.35382
Summary: We investigate a class of impulsive Cohen-Grossberg neural networks with distributed delays and reaction-diffusion terms. By establishing an integro-differential inequality with impulsive initial conditions and applying M-matrix theory, we find some sufficient conditions ensuring the existence, uniqueness, global exponential stability and global robust exponential stability of equilibrium point for impulsive Cohen-Grossberg neural networks with distributed delays and reaction-diffusion terms. An example is given to illustrate the results obtained here.
MSC:
35K57Reaction-diffusion equations
35B35Stability of solutions of PDE
35K20Second order parabolic equations, initial boundary value problems
35R10Partial functional-differential equations
37N25Dynamical systems in biology
92B20General theory of neural networks (mathematical biology)
References:
[1]Cohen, M.; Grossberg, S.: Absolute stability and global pattern formation and parallel memory storage by competitive neural networks, IEEE trans. Syst. man cybern. 13, 815-826 (1983) · Zbl 0553.92009
[2]Yang, Z.; Xu, D.: Impulsive effects on stability of Cohen – Grossberg neural networks with variable delays, Appl. math. Comput. 177, 63-78 (2006) · Zbl 1103.34067 · doi:10.1016/j.amc.2005.10.032
[3]Chen, Z.; Ruan, J.: Global stability analysis of impulsive Cohen – Grossberg neural networks with delay, Phys. lett. A 345, 101-111 (2005)
[4]Chen, Z.; Ruan, J.: Global dynamic analysis of general Cohen – Grossberg neural networks with impulse, Chaos soliton. Fract. 32, 1830-1837 (2007) · Zbl 1142.34045 · doi:10.1016/j.chaos.2005.12.018
[5]Cao, J.; Liang, J.: Boundedness and stability for Cohen – Grossberg neural network with time-varying delays, J. math. Anal. appl. 296, 665-685 (2004) · Zbl 1044.92001 · doi:10.1016/j.jmaa.2004.04.039
[6]Yuan, K.; Cao, J.: An analysis of global asymptotic stability of delayed Cohen – Grossberg neural networks via nonsmooth analysis, IEEE trans. Circuit. syst. I 52, No. 9, 1854-1861 (2005)
[7]Arik, S.; Orman, Z.: Global stability analysis of Cohen – Grossberg neural networks with time varying delays, Phys. lett. A 341, 410-421 (2005) · Zbl 1171.37337 · doi:10.1016/j.physleta.2005.04.095
[8]Zhang, J.; Suda, Y.; Komine, H.: Global exponential stability of Cohen – Grossberg neural networks with variable delays, Phys. lett. A 338, 44-55 (2005) · Zbl 1136.34347 · doi:10.1016/j.physleta.2005.02.005
[9]Liao, X.; Li, C.: Global attractivity of Cohen – Grossberg model with finite and infinite delays, J. math. Anal. appl. 315, 244-262 (2006) · Zbl 1098.34062 · doi:10.1016/j.jmaa.2005.04.076
[10]Jiang, M.; Shen, Y.; Liao, X.: Boundedness and global exponential stability for generalized Cohen – Grossberg neural networks with variable delay, Appl. math. Comput. 172, 379-393 (2006) · Zbl 1090.92004 · doi:10.1016/j.amc.2005.02.009
[11]Wan, A.; Qiao, H.; Peng, J.; Wang, M.: Delay-independent criteria for exponential stability of generalized Cohen – Grossberg neural networks with discrete delays, Phys. lett. A 353, 151-157 (2006)
[12]Song, Q.; Cao, J.: Stability analysis of Cohen – Grossberg neural network with both time-varying and continuously distributed delays, J. comput. Appl. math. 197, 188-203 (2006) · Zbl 1108.34060 · doi:10.1016/j.cam.2005.10.029
[13]Zhao, H.; Wang, K.: Dynamical behaviors of Cohen – Grossberg neural networks with delays and reaction – diffusion terms, Neurocomputing 70, 536-543 (2006)
[14]Li, K.; Li, Z.; Zhang, X.: Exponential stability of reaction – diffusion generalized Cohen – Grossberg neural networks with both variable and distributed delays, Int. math. Forum 2, 1397-1414 (2007) · Zbl 1139.92001
[15]Song, Q.; Cao, J.; Zhao, Z.: Periodic solutions and its exponential stability of reaction – diffusion recurrent neural networks with continuously distributed delays, Nonlinear anal.: real world appl. 7, 65-80 (2006) · Zbl 1094.35128 · doi:10.1016/j.nonrwa.2005.01.004
[16]Zhao, Z.; Song, Q.; Zhang, J.: Exponential periodicity and stability of neural networks with reaction – diffusion terms and both variable and unbounded delays, Comput. math. Appl. 51, 475-486 (2006) · Zbl 1104.35065 · doi:10.1016/j.camwa.2005.10.009
[17]Li, Y.; Yang, C.: Global exponential stability analysis on impulsive BAM neural networks with distributed delays, J. math. Anal. appl. 324, 1125-1139 (2006) · Zbl 1102.68117 · doi:10.1016/j.jmaa.2006.01.016
[18]Liang, J.; Cao, J.: Global exponential stability of reaction – diffusion recurrent neural networks with timevarying delays, Phys. lett. A 314, 434-442 (2003) · Zbl 1052.82023 · doi:10.1016/S0375-9601(03)00945-9
[19]Wang, L.; Xu, D.: Global exponential stability of Hopfield reaction – diffusion neural networks with variable delays, Sci. China ser. F 46, 466-474 (2003) · Zbl 1186.82062 · doi:10.1360/02yf0146
[20]Lou, X.; Cui, B.: Boundedness and exponential stability for nonautonomous cellular neural networks with reaction – diffusion terms, Chaos soliton. Fract. 33, 653-662 (2007) · Zbl 1133.35386 · doi:10.1016/j.chaos.2006.01.044
[21]Song, Q.; Cao, J.: Global exponential stability and existence of periodic solutions in BAM networks with delays and reaction – diffusion terms, Chaos soliton. Fract. 23, 421-430 (2005) · Zbl 1068.94534 · doi:10.1016/j.chaos.2004.04.011
[22]Allegretto, W.; Papini, D.: Stability for delayed reaction – diffusion neural networks, Phys. lett. A 360, 669-680 (2007)
[23]Zhou, Q.; Wan, L.; Sun, J.: Exponential stability of reaction – diffusion generalized Cohen – Grossberg neural networks with time-varying delays, Chaos soliton. Fract. 32, 1713-1719 (2007) · Zbl 1132.35309 · doi:10.1016/j.chaos.2005.12.003
[24]Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations, (1989) · Zbl 0718.34011
[25]Gopalsamy, K.: Stability of artificial neural networks with impulses, Appl. math. Comput. 154, 783-813 (2004) · Zbl 1058.34008 · doi:10.1016/S0096-3003(03)00750-1
[26]Guan, Z.; Chen, G.: On delayed impulsive Hopfield neural networks, Neural networks 12, 273-280 (1999)
[27]Guan, Z.; James, L.; Chen, G.: On impulsive auto-associative neural networks, Neural networks 13, 63-69 (2000)
[28]Akca, H.; Alassar, R.; Covachev, V.; Covacheva, Z.; Al-Zahrani, E.: Continuous-time additive Hopfield-type neural networks with impulses, J. math. Anal. appl. 290, 436-451 (2004) · Zbl 1057.68083 · doi:10.1016/j.jmaa.2003.10.005
[29]Li, Y.; Lu, L.: Global exponential stability and existence of periodic solution of Hopfield-type neural networks with impulses, Phys. lett. A 333, 62-71 (2004) · Zbl 1123.34303 · doi:10.1016/j.physleta.2004.09.083
[30]Li, Y.: Global exponential stability of BAM neural networks with delays and impulses, Chaos solition. Fract. 24, 279-285 (2005) · Zbl 1099.68085 · doi:10.1016/j.chaos.2004.09.027
[31]Li, Y.; Xing, W.; Lu, L.: Existence and global exponential stability of periodic solution of a class of neural networks with impulses, Chaos soliton. Fract. 27, 437-445 (2006) · Zbl 1084.68103 · doi:10.1016/j.chaos.2005.04.021
[32]Yang, Y.; Cao, J.: Stability and periodicity in delayed cellular neural networks with impulsive effects, Nonlinear anal.: real world appl. 8, 362-374 (2007) · Zbl 1115.34072 · doi:10.1016/j.nonrwa.2005.11.004
[33]Qiu, J.: Exponential stability of impulsive neural networks with time-varying delays and reaction – diffusion terms, Neurocomputing 70, 1102-1108 (2007)
[34]Xu, D.; Yang, Z.: Impulsive delay differential inequality and stability of neural networks, J. math. Anal. appl. 305, 107-120 (2005) · Zbl 1091.34046 · doi:10.1016/j.jmaa.2004.10.040
[35]Akhmet, M. U.; Beklioglu, M.; Ergenc, T.; Tkachenko, V. I.: An impulsive ratio-dependent predatorcprey system with diffusion, Nonlinear anal.: real world appl. 7, 1255-1267 (2006) · Zbl 1114.35097 · doi:10.1016/j.nonrwa.2005.11.007
[36]Berman, A.; Plemmons, R. J.: Nonnegative matrices in mathematical sciences, (1979) · Zbl 0484.15016
[37]Gopalsamy, K.; He, X.: Delay-independent stabilily in bi-directional associative menury neural networks, IEEE trans. Neural networks 5, 998-1002 (1994)