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May  2014, 19(3): 761-773. doi: 10.3934/dcdsb.2014.19.761

Average criteria for periodic neural networks with delay

Benedetta Lisena 1,

1. 

Dipartimento di Matematica, Universitá degli studi di Bari, 70125 Bari, Italy

Received  June 2013 Revised  October 2013 Published  February 2014

By using Lyapunov functions and some recent estimates of Halanay type, new criteria are introduced for the global exponential stability of a class of cellular neural networks, with delay and periodic coefficients and inputs. The novelty of those criteria lies in the fact that they are very efficient in presence of oscillating coefficients, because they are given in average form.
Keywords: exponential stability, delay, Halanay inequality, average criteria., Neural networks.
Mathematics Subject Classification: Primary: 34K13, 34K20; Secondary: 34K6.
Citation: Benedetta Lisena. Average criteria for periodic neural networks with delay. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 761-773. doi: 10.3934/dcdsb.2014.19.761
References:
[1]

S. Ahmad and I. M. Stamova, Global exponential stability for impulsive cellular neural networks with time-delays, Nonlinear Anal., 69 (2008), 786-795. doi: 10.1016/j.na.2008.02.067.  Google Scholar

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B. Lisena, Dynamical behavior of impulsive and periodic Cohen-Grossberg neural networks, Nonlinear Anal., 74 (2011), 4511-4519. doi: 10.1016/j.na.2011.04.015.  Google Scholar

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B. Lisena, Asymptotic properties in a delay differential inequality with periodic coefficients, Mediterr. J. Math., 10 (2013), 1717-1730. doi: 10.1007/s00009-013-0261-5.  Google Scholar

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B. Liu and L. Huang, Existence and exponential stability of periodic solutions for cellular neural networks with time-varying delays, Phys. Lett. A, 349 (2006), 474-483.  Google Scholar

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H. Liu and L. Wang, Globally exponential stability and periodic solutions of CNNs with variable coefficients and variable delays, Chaos Solitons Fractals, 29 (2006), 1137-1141. doi: 10.1016/j.chaos.2005.08.120.  Google Scholar

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S. Long and D. Xu, Delay-dependent stability analysis for impulsive neural networks with time varying delays, Neurocomputing, 71 (2008), 1705-1713. doi: 10.1016/j.neucom.2007.03.010.  Google Scholar

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S. Mohamad and K. Gopalsamy, Exponential stability of continuous-time and discrete-time cellular neural networks with delays, Appl. Math. Comput., 135 (2003), 17-38. doi: 10.1016/S0096-3003(01)00299-5.  Google Scholar

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Y. Shao, Exponential stability of periodic neural networks with impulsive effects and time-varying delays, Appl. Math. Comput., 217 (2011), 6893-6899. doi: 10.1016/j.amc.2011.01.068.  Google Scholar

[13]

I. M. Stamova and R. Ilarionov, On global exponential stability for impulsive cellular neural networks with time-varying delays, Comput. Math. Appl., 59 (2010), 3508-3515. doi: 10.1016/j.camwa.2010.03.043.  Google Scholar

[14]

M. Tan and Y. Tan, Global exponential stability of periodic solution of neural network with variable coefficients and time-varying delays, Appl. Math. Model., 33 (2009), 373-385. doi: 10.1016/j.apm.2007.11.010.  Google Scholar

[15]

H. Wang, C. Li and H. Xu, Existence and global exponential stability of periodic solution of cellular neural networks with delay and impulses, Results Math., 58 (2010), 191-204. doi: 10.1007/s00025-010-0048-y.  Google Scholar

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show all references

References:
[1]

S. Ahmad and I. M. Stamova, Global exponential stability for impulsive cellular neural networks with time-delays, Nonlinear Anal., 69 (2008), 786-795. doi: 10.1016/j.na.2008.02.067.  Google Scholar

[2]

H. Gu, H. Jiang and Z. Teng, Stability and periodicity in high-order neural networks with impulsive effects, Nonlinear Anal., 68 (2008), 3186-3200. doi: 10.1016/j.na.2007.03.024.  Google Scholar

[3]

H. Jiang, Z. Li and Z. Teng, Boundedness and stability for nonautonomous cellular networks with delays, Phys. Lett. A, 306 (2003), 313-325. doi: 10.1016/S0375-9601(02)01608-0.  Google Scholar

[4]

B. Li and D. Xu, Existence and exponential stability of periodic solution for impulsive Cohen-Grossberg neural networks with time varying delays, Appl. Math. Comput., 219 (2012), 2506-2520. doi: 10.1016/j.amc.2012.08.086.  Google Scholar

[5]

B. Lisena, Exponential stability of Hopfield neural networks with impulses, Nonlinear Anal. Real World Appl., 12 (2011), 1923-1930. doi: 10.1016/j.nonrwa.2010.12.008.  Google Scholar

[6]

B. Lisena, Dynamical behavior of impulsive and periodic Cohen-Grossberg neural networks, Nonlinear Anal., 74 (2011), 4511-4519. doi: 10.1016/j.na.2011.04.015.  Google Scholar

[7]

B. Lisena, Asymptotic properties in a delay differential inequality with periodic coefficients, Mediterr. J. Math., 10 (2013), 1717-1730. doi: 10.1007/s00009-013-0261-5.  Google Scholar

[8]

B. Liu and L. Huang, Existence and exponential stability of periodic solutions for cellular neural networks with time-varying delays, Phys. Lett. A, 349 (2006), 474-483.  Google Scholar

[9]

H. Liu and L. Wang, Globally exponential stability and periodic solutions of CNNs with variable coefficients and variable delays, Chaos Solitons Fractals, 29 (2006), 1137-1141. doi: 10.1016/j.chaos.2005.08.120.  Google Scholar

[10]

S. Long and D. Xu, Delay-dependent stability analysis for impulsive neural networks with time varying delays, Neurocomputing, 71 (2008), 1705-1713. doi: 10.1016/j.neucom.2007.03.010.  Google Scholar

[11]

S. Mohamad and K. Gopalsamy, Exponential stability of continuous-time and discrete-time cellular neural networks with delays, Appl. Math. Comput., 135 (2003), 17-38. doi: 10.1016/S0096-3003(01)00299-5.  Google Scholar

[12]

Y. Shao, Exponential stability of periodic neural networks with impulsive effects and time-varying delays, Appl. Math. Comput., 217 (2011), 6893-6899. doi: 10.1016/j.amc.2011.01.068.  Google Scholar

[13]

I. M. Stamova and R. Ilarionov, On global exponential stability for impulsive cellular neural networks with time-varying delays, Comput. Math. Appl., 59 (2010), 3508-3515. doi: 10.1016/j.camwa.2010.03.043.  Google Scholar

[14]

M. Tan and Y. Tan, Global exponential stability of periodic solution of neural network with variable coefficients and time-varying delays, Appl. Math. Model., 33 (2009), 373-385. doi: 10.1016/j.apm.2007.11.010.  Google Scholar

[15]

H. Wang, C. Li and H. Xu, Existence and global exponential stability of periodic solution of cellular neural networks with delay and impulses, Results Math., 58 (2010), 191-204. doi: 10.1007/s00025-010-0048-y.  Google Scholar

[16]

Z. Yuan and L. Yuan, Existence and global convergence of periodic solution of delayed neural networks, Math. Comput. Modelling, 48 (2008), 101-113. doi: 10.1016/j.mcm.2007.08.010.  Google Scholar

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