zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Stability analysis for impulsive Cohen-Grossberg neural networks with time-varying delays and distributed delays. (English) Zbl 1162.92002
Summary: A class of impulsive {\it M. A. Cohen} and {\it S. Grossberg} [IEEE Trans. Syst. Man. Cybern. 13, 815--826 (1983; Zbl 0553.92009)] neural networks with time-varying delays and distributed delays is investigated. By establishing an integro-differential inequality with impulsive initial conditions, employing $M$-matrix theory and a nonlinear measure approach, some new sufficient conditions ensuring the existence, uniqueness, global exponential stability and global robust exponential stability of equilibrium points for impulsive Cohen-Grossberg neural networks with time-varying delays and distributed delays are obtained. In particular, a more precise estimate of the exponential convergence rate is provided. By comparisons and examples, it is shown that the results obtained here can extremely extend and improve previously known results.

MSC:
92B20General theory of neural networks (mathematical biology)
34K20Stability theory of functional-differential equations
34K45Functional-differential equations with impulses
34K60Qualitative investigation and simulation of models
68T05Learning and adaptive systems
WorldCat.org
Full Text: DOI
References:
[1] Akça, 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
[2] 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
[3] Berman, A.; Plemmons, R. J.: Nonnegative matrices in mathematical sciences. (1979) · Zbl 0484.15016
[4] 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
[5] Chen, Z.; Ruan, J.: Global stability analysis of impulsive Cohen--Grossberg neural networks with delay. Phys. lett. A 345, 101-111 (2005) · Zbl 05314183
[6] Chen, Z.; Ruan, J.: Global dynamic analysis of general Cohen--Grossberg neural networks with impulse. Chaos solitons fractals 32, 1830-1837 (2007) · Zbl 1142.34045
[7] 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
[8] Gopalsamy, K.; He, X.: Delay-independent stability in bidirectional associative memory networks. IEEE trans. Neural networks 5, 998-1002 (1994)
[9] Gopalsamy, K.: Stability of artificial neural networks with impulses. Appl. math. Comput. 154, 783-813 (2004) · Zbl 1058.34008
[10] Guan, Z.; Chen, G.: On delayed impulsive Hopfield neural networks. Neural networks 12, 273-280 (1999)
[11] Guan, Z.; James, L.; Chen, G.: On impulsive auto-associative neural networks. Neural networks 13, 63-69 (2000)
[12] Huang, T.; Chan, A.; Huang, Y.; Cao, J.: Stability of Cohen--Grossberg neural networks with time-varying delays. Neural networks 20, 868-873 (2007) · Zbl 1151.68583
[13] Huang, T.; Li, C.; Chen, G.: Stability of Cohen--Grossberg neural networks with unbounded distributed delays. Chaos, solitons and fractals 34, 992-996 (2007) · Zbl 1137.34351
[14] Huang, Z.; Yang, Q.; Luo, X.: Exponential stability of impulsive neural networks with time-varying delays. Chaos, solitons and fractals 35, 770-780 (2008) · Zbl 1139.93353
[15] 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
[16] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations. (1989) · Zbl 0719.34002
[17] Li, Y.: Global exponential stability of BAM neural networks with delays and impulses. Chaos, solitons & fractals 24, 279-285 (2005) · Zbl 1099.68085
[18] 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
[19] 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
[20] C. Li, S. Yang, Global attractivity in delayed Cohen--Grossberg neural network models, Chaos Solitons Fractals, in press (doi:10.1016/j.chaos.2007.06.064) · Zbl 1197.34137
[21] Li, Y.; Xing, W.; Lu, L.: Existence and global exponential stability of periodic solution of a class of neural networks with impulses. Chaos solitons & fractals 27, 437-445 (2006) · Zbl 1084.68103
[22] Liao, X.; Wong, K. -W.; Li, C.: Global exponential stability for a class of generalized neural networks with distributed delays. Nonlinear anal. RWA 5, 527-547 (2004) · Zbl 1094.34053
[23] Liao, X.; Li, C.; Wong, K. -W.: Criteria for exponential stability of Cohen--Grossberg neural networks. Neural networks 17, 1401-1414 (2004) · Zbl 1073.68073
[24] 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
[25] Liang, J.; Cao, J.: Global asymptotic stability of bi-directional associative memory networks with distributed delays. Appl. math. Comput. 152, 415-424 (2004) · Zbl 1046.94020
[26] Mohamad, S.; Gopalsamy, K.; Akça, H.: Exponential stability of artificial neural networks with distributed delays and large impulses. Nonlinear anal. RWA 9, 872-888 (2008) · Zbl 1154.34042
[27] Park, Ju H.; Cho, H. J.: A delay-dependent asymptotic stability criterion of cellular neural networks with time-varying discrete and distributed delays. Chaos solitons fractals 33, 436-442 (2007) · Zbl 1142.34379
[28] Song, Q.; Cao, J.: Stability analysis of Cohen--Grossberg neural network with both time-varying and continuously distributed delays. J. comp. Appl. math. 197, 188-203 (2006) · Zbl 1108.34060
[29] Song, Q.; Zhang, J.: Global exponential stability of impulsive Cohen--Grossberg neural network with time-varying delays. Nonlinear anal. RWA 9, 500-510 (2008) · Zbl 1142.34046
[30] 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) · Zbl 05675810
[31] Wan, L.; Sun, J.: Global asymptotic stability of Cohen--Grossberg neural network with continuously distributed delays. Phys. lett. A 342, 331-340 (2005) · Zbl 1222.93200
[32] Wang, Y.; Xiong, W.; Zhou, Q.; Xiao, B.; Yu, Y.: Global exponential stability of cellular neural networks with continuously distributed delays and impulses. Phys. lett. A 350, 89-95 (2006) · Zbl 1195.34064
[33] Wang, L.: Stability of Cohen--Grossberg neural networks with distributed delays. Appl. math. Comput. 160, 93-110 (2005) · Zbl 1069.34113
[34] Xia, Y.; Cao, J.; Cheng, S.: Global exponential stability of delayed cellular neural networks with impulses. Neurocomputing 70, 2495-2501 (2007)
[35] Xiong, W.; Zhou, Q.; Xiao, B.; Yu, Y.: Global exponential stability of cellular neural networks with mixed delays and impulses. Chaos, solitons and fractals 34, 896-902 (2007) · Zbl 1137.34358
[36] Xu, D.; Yang, Z.: Impulsive delay differential inequality and stability of neural networks. J. math. Anal. appl. 305, 107-120 (2005) · Zbl 1091.34046
[37] Xu, D.; Zhu, W.; Long, S.: Global exponential stability of impulsive integro-differential equation. Nonlinear anal. 64, 2805-2816 (2006) · Zbl 1093.45004
[38] 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
[39] Yang, Y.; Cao, J.: Stability and periodicity in delayed cellular neural networks with impulsive effects. Nonlinear anal. RWA 8, 362-374 (2007) · Zbl 1115.34072
[40] Yuan, K.; Cao, J.: An analysis of global asymptotic stability of delayed Cohen--Grossberg neural networks via nonsmooth analysis. IEEE trans. Circuits syst. I 52, No. 9, 1854-1861 (2005)
[41] Zhang, J.: Global exponential stability of interval neural networks with variable delays. Appl. math. Lett. 19, 1222-1227 (2006) · Zbl 1180.34083
[42] Zhang, J.; Suda, Y.; Iwasa, T.: Absolutely exponential stability of a class of neural networks with unbounded delay. Neural networks 17, 391-397 (2004) · Zbl 1074.68057
[43] 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
[44] Zhang, Q.; Wei, X.; Xu, J.: Global exponential stability of Hopfield neural networks with continuously distributed delays. Phys. lett. A 315, 431-436 (2003) · Zbl 1038.92002
[45] Zhao, H.: Global asymptotic stability of Hopfield neural network involving distributed delays. Neural networks 17, 47-53 (2004) · Zbl 1082.68100