\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

On the stability of sets for reaction–diffusion Cohen–Grossberg delayed neural networks

  • * Corresponding author: Ivanka Stamova

    * Corresponding author: Ivanka Stamova 
This paper is supported in part by the European Regional Development Fund through the Operational Programme "Science and Education for Smart Growth" under Contract UNITe No. BG05M2OP001–1.001–0004 (2018–2023)
Abstract / Introduction Full Text(HTML) Related Papers Cited by
  • In this paper, we introduce the notion of stability of sets for reaction-diffusion Cohen–Grossberg neural networks with time-varying delays. The Lyapunov–Razumikhin technique and a comparison principle are adapted to prove the new stability criteria. In addition, the obtained results are extended to the uncertain case, and the robust stability notion is also investigated. Examples are considered to demonstrate the effectiveness of our results.

    Mathematics Subject Classification: Primary: 92B20, 34K20; Secondary: 34K60.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] C. Aouiti and E. A. Assali, Nonlinear Lipschitz measure and adaptive control for stability and synchronization in delayed inertial Cohen-Grossberg-type neural networks, Int. J. Adapt. Control, 33 (2019), 1457-1477.  doi: 10.1002/acs.3042.
    [2] S. Baldwin and E. E. Slaminka, A stable/unstable "manifold" theorem for area preserving homeomorphisms of two manifolds, Proc. Amer. Math. Soc., 109 (1990), 823-828.  doi: 10.2307/2048225.
    [3] M. Bohner, G. Tr. Stamov and I. M. Stamova, Almost periodic solutions of Cohen–Grossberg neural networks with time-varying delay and variable impulsive perturbations, Commun. Nonlinear Sci. Numer. Simul., 80 (2020), 104952, 14pp. doi: 10.1016/j.cnsns.2019.104952.
    [4] W. H. ChenL. Liu and X. Lu, Intermittent synchronization of reaction-diffusion neural networks with mixed delays via Razumikhin technique, Nonlinear Dyn., 87 (2017), 535-551. 
    [5] M. A. Cohen and S. Grossberg, Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE Trans. Syst. Man Cybern., 13 (1983), 815-826. 
    [6] Q. Gan, Adaptive synchronization of Cohen-Grossberg neural networks with unknown parameters and mixed time-varying delays, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 3040-3049.  doi: 10.1016/j.cnsns.2011.11.012.
    [7] Q. Gan, Exponential synchronization of generalized neural networks with mixed time-varying delays and reaction-diffusion terms via aperiodically intermittent control, Chaos, 27 (2017), 013113, 10pp. doi: 10.1063/1.4973976.
    [8] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, 1$^st$ edition, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.
    [9] Y.-G. KaoJ.-F. GuoC.-H. Wang and X.-Q. Sun, Delay-dependent robust exponential stability of Markovian jumping reaction-diffusion Cohen–Grossberg neural networks with mixed delays, J. Franklin Inst., 349 (2012), 1972-1988.  doi: 10.1016/j.jfranklin.2012.04.005.
    [10] Y. KaoC. Wang and L. Zhang, Delay-dependent robust exponential stability of impulsive Markovian jumping reaction-diffusion Cohen–Grossberg neural networks, Neural Process. Lett., 38 (2013), 321-346. 
    [11] H. K. Khalil, Nonlinear Systems, 3$^{rd}$ edition, Prentice-Hall, Upper Saddle River, NJ, 2002.
    [12] X. Li, Existence and global exponential stability of periodic solution for impulsive Cohen–Grossberg-type BAM neural networks with continuously distributed delays, Appl. Math. Comput., 215 (2009), 292-307.  doi: 10.1016/j.amc.2009.05.005.
    [13] X. Li, Exponential stability of Cohen–Grossberg-type BAM neural networks with time-varying delays via impulsive control, Neurocomputing, 73 (2009), 525-530. 
    [14] X. LiJ. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Appl. Math. Comput., 329 (2018), 14-22.  doi: 10.1016/j.amc.2018.01.036.
    [15] X. LiX. Yang and T. Huang, Persistence of delayed cooperative models: Impulsive control method, Appl. Math. Comput., 342 (2019), 130-146.  doi: 10.1016/j.amc.2018.09.003.
    [16] R. Li and J. Cao, Stability analysis of reaction-diffusion uncertain memristive neural networks with time-varying delays and leakage term, Appl. Math. Comput., 278 (2016), 54-69.  doi: 10.1016/j.amc.2016.01.016.
    [17] R. LiJ. CaoA. Alsaedi and B. Ahmad, Passivity analysis of delayed reaction-diffusion Cohen–Grossberg neural networks via Hardy–Poincarè inequality, J. Franklin Inst., 354 (2017), 3021-3038.  doi: 10.1016/j.jfranklin.2017.02.028.
    [18] L. Li and J. Jian, Exponential convergence and Lagrange stability for impulsive Cohen–Grossberg neural networks with time-varying delays, J. Comput. Appl. Math., 277 (2015), 23-35.  doi: 10.1016/j.cam.2014.08.029.
    [19] B. LiuX. Liu and X. Liao, Robust stability of uncertain impulsive dynamical systems, J. Math. Anal. Appl., 290 (2004), 519-533.  doi: 10.1016/j.jmaa.2003.10.035.
    [20] X. Lou and B. Cui, Boundedness and exponential stability for nonautonomous cellular neural networks with reaction-diffusion terms, Chaos Solitons Fractals, 33 (2007), 653-662.  doi: 10.1016/j.chaos.2006.01.044.
    [21] J. G. Lu, Global exponential stability and periodicity of reaction-diffusion delayed recurrent neural networks with Dirichlet boundary conditions, Chaos Solitons Fractals, 35 (2008), 116-125.  doi: 10.1016/j.chaos.2007.05.002.
    [22] F. M. G. Magpantay and A. R. Humphries, Generalized Lyapunov–Razumikhin techniques for scalar state-dependent delay differential equations, Discrete Continuous Dynam. Systems - S, 13 (2020), 85-104.  doi: 10.3934/dcdss.2020005.
    [23] J. PanX. Liu and S. Zhong, Stability criteria for impulsive reaction-diffusion Cohen–Grossberg neural networks with time-varying delays, Math. Comput. Modelling, 51 (2010), 1037-1050.  doi: 10.1016/j.mcm.2009.12.004.
    [24] J. Pan and S. Zhong, Dynamical behaviors of impulsive reaction-diffusion Cohen–Grossberg neural network with delays, Neurocomputing, 73 (2010), 1344-1351. 
    [25] A. PratapR. RajaJ. CaoC. P. Lim and O. Bagdasar, Stability and pinning synchronization analysis of fractional order delayed Cohen-Grossberg neural networks with discontinuous activations, Appl. Math. Comput., 359 (2019), 241-260.  doi: 10.1016/j.amc.2019.04.062.
    [26] J. Qiu, Exponential stability of impulsive neural networks with time-varying delays and reaction-diffusion terms, Neurocomputing, 70 (2007), 1102-1108. 
    [27] R. RakkiyappanS. Dharani and Q. Zhu, Synchronization of reaction-diffusion neural networks with time-varying delays via stochastic sampled-data controller, Nonlinear Dyn., 79 (2015), 485-500.  doi: 10.1007/s11071-014-1681-x.
    [28] R. Rao, Delay-dependent exponential stability for nonlinear reaction-diffusion uncertain Cohen–Grossberg neural networks with partially known transition rates via Hardy–Poincarè inequality, Chin. Ann. Math., 35 (2014), 575-598.  doi: 10.1007/s11401-014-0839-7.
    [29] F. R. Ruiz del Portal, Stable sets of planar homeomorphisms with translation pseudo-arcs, Discrete Continuous Dynam. Systems - S, 12 (2019), 2379-2390.  doi: 10.3934/dcdss.2019149.
    [30] R. SkjetneT. I. Fossen and P. V. Kokotovic, Adaptive maneuvering, with experiments, for a model ship in a marine control laboratory, Automatica, 41 (2005), 289-298.  doi: 10.1016/j.automatica.2004.10.006.
    [31] I. M. Stamova, Stability Analysis of Impulsive Functional Differential Equations, 1$^st$ edition, Walter de Gruyter, Berlin, 2009. doi: 10.1515/9783110221824.
    [32] I. M. Stamova and S. Simeonov, Delayed reaction-diffusion cellular neural networks of fractional order: Mittag–Leffler stability and synchronization, J. Comput. Nonlinear Dynam., 13 (2018), 011015 (7 pages). doi: 10.1115/1.4038290.
    [33] I. M. Stamova and G. T. Stamov, On the stability of sets for delayed Kolmogorov-type systems, Proc. Amer. Math. Soc., 142 (2014), 591-601.  doi: 10.1090/S0002-9939-2013-12197-0.
    [34] I. M. Stamova and G. T. Stamov, Applied Impulsive Mathematical Models, 1$^{st}$ edition, Springer (CMS Books in Mathematics), Cham, 2016. doi: 10.1007/978-3-319-28061-5.
    [35] I. Stamova and G. Stamov, Mittag–Leffler synchronization of fractional neural networks with time-varying delays and reaction-diffusion terms using impulsive and linear controllers, Neural Netw., 96 (2017), 22-32.  doi: 10.1016/j.neunet.2017.08.009.
    [36] C. Vidhya and P. Balasubramaniam, Robust stability of uncertain Markovian jumping stochastic Cohen–Grossberg type BAM neural networks with time-varying delays and reaction diffusion terms, Neural Parallel Sci. Comput., 19 (2011), 181-195. 
    [37] Y. WanJ. CaoG. Wen and W. Yu, Robust fixed–ime synchronization of delayed Cohen–Grossberg neural networks, Neural Netw., 73 (2016), 86-94. 
    [38] J.-L. WangH.-N. Wu and L. Guo, Stability analysis of reaction-diffusion Cohen–Grossberg neural networks under impulsive control, Neurocomputing, 106 (2013), 21-30.  doi: 10.1016/j.neucom.2012.11.006.
    [39] Z. Wang and H. Zhang, Global asymptotic stability of reaction-diffusion Cohen–Grossberg neural networks with continuously distributed delays, IEEE Trans. Neral Netw., 21 (2010), 39-49. 
    [40] P. Yan and T. Lv, Periodicity of delayed reaction-diffusion high-order Cohen–Grossberg neural networks with Dirichlet boundary conditions, Rocky Mountain J. Math., 41 (2011), 949-970.  doi: 10.1216/RMJ-2011-41-3-949.
    [41] X. YangJ. Cao and Z. Yang, Synchronization of coupled reaction-diffusion neural networks with time-varying delays via pinning-impulsive control, SIAM J. Control Opt., 51 (2013), 3486-3510.  doi: 10.1137/120897341.
    [42] D. YangX. Li and J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Anal. - Hybri. Syst., 32 (2019), 294-305.  doi: 10.1016/j.nahs.2019.01.006.
    [43] T. Yoshizawa, Stability Theory by Liapunov's Second Method, 1$^{st}$ edition, The Mathematical Society of Japan, Tokyo, 1966.
    [44] K. YuanJ. Cao and H. Li, Robust stability of switched Cohen–Grossberg neural networks with mixed time-varying delays, IEEE Trans. Syst. Man Cybern., 36 (2006), 1356-1363. 
    [45] A. I. Zecevic and D. D. Siljak, Control of Complex Systems: Structural Constraints and Uncertainty, 1$^{st}$ edition, Springer, New York, 2010.
    [46] Q. Zhu and X. Li, Exponential and almost sure exponential stability of stochastic fuzzy delayed Cohen–Grossberg neural networks, Fuzzy Sets Syst., 203 (2012), 74-94.  doi: 10.1016/j.fss.2012.01.005.
  • 加载中
SHARE

Article Metrics

HTML views(1713) PDF downloads(315) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return