# American Institute of Mathematical Sciences

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

## Average criteria for periodic neural networks with delay

 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.
Citation: Benedetta Lisena. Average criteria for periodic neural networks with delay. Discrete and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
 [1] Sylvia Novo, Rafael Obaya, Ana M. Sanz. Exponential stability in non-autonomous delayed equations with applications to neural networks. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 517-536. doi: 10.3934/dcds.2007.18.517 [2] Muhammet Mert Ketencigil, Ozlem Faydasicok, Sabri Arik. Novel criteria for robust stability of Cohen-Grossberg neural networks with multiple time delays. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022081 [3] Ling Zhang, Xiaoqi Sun. Stability analysis of time-varying delay neural network for convex quadratic programming with equality constraints and inequality constraints. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022035 [4] Pham Huu Anh Ngoc. New criteria for exponential stability in mean square of stochastic functional differential equations with infinite delay. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021040 [5] Quan Hai, Shutang Liu. Mean-square delay-distribution-dependent exponential synchronization of chaotic neural networks with mixed random time-varying delays and restricted disturbances. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3097-3118. doi: 10.3934/dcdsb.2020221 [6] Yong Ren, Huijin Yang, Wensheng Yin. Weighted exponential stability of stochastic coupled systems on networks with delay driven by $G$-Brownian motion. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3379-3393. doi: 10.3934/dcdsb.2018325 [7] Ricai Luo, Honglei Xu, Wu-Sheng Wang, Jie Sun, Wei Xu. A weak condition for global stability of delayed neural networks. Journal of Industrial and Management Optimization, 2016, 12 (2) : 505-514. doi: 10.3934/jimo.2016.12.505 [8] Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219 [9] Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577 [10] Xiaochen Mao, Weijie Ding, Xiangyu Zhou, Song Wang, Xingyong Li. Complexity in time-delay networks of multiple interacting neural groups. Electronic Research Archive, 2021, 29 (5) : 2973-2985. doi: 10.3934/era.2021022 [11] Pierdomenico Pepe. A nonlinear version of Halanay's inequality for the uniform convergence to the origin. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021045 [12] Ivanka Stamova, Gani Stamov. On the stability of sets for reaction–diffusion Cohen–Grossberg delayed neural networks. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1429-1446. doi: 10.3934/dcdss.2020370 [13] István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773 [14] Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367 [15] István Győri, László Horváth. Sharp estimation for the solutions of delay differential and Halanay type inequalities. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3211-3242. doi: 10.3934/dcds.2017137 [16] Yong Ren, Wensheng Yin, Dongjin Zhu. Exponential stability of SDEs driven by $G$-Brownian motion with delayed impulsive effects: average impulsive interval approach. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3347-3360. doi: 10.3934/dcdsb.2018248 [17] Jianping Zhou, Yamin Liu, Ju H. Park, Qingkai Kong, Zhen Wang. Fault-tolerant anti-synchronization control for chaotic switched neural networks with time delay and reaction diffusion. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1569-1589. doi: 10.3934/dcdss.2020357 [18] Alaa Hayek, Serge Nicaise, Zaynab Salloum, Ali Wehbe. Exponential and polynomial stability results for networks of elastic and thermo-elastic rods. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1183-1220. doi: 10.3934/dcdss.2021142 [19] Ying Sue Huang. Resynchronization of delayed neural networks. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 397-401. doi: 10.3934/dcds.2001.7.397 [20] Zhijian Yang, Yanan Li. Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous kirchhoff wave models. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2629-2653. doi: 10.3934/dcds.2018111

2020 Impact Factor: 1.327