# American Institute of Mathematical Sciences

## Mean-square delay-distribution-dependent exponential synchronization of chaotic neural networks with mixed random time-varying delays and restricted disturbances

 1 College of Control Science and Engineering, Shandong University, Jinan 250061, China 2 College of Mathematics Science Inner Mongolia Normal University Hohhot 010022, China

* Corresponding author: stliu618@163.com

Received  December 2019 Revised  April 2020 Published  July 2020

Fund Project: The first author is supported by NSF of china 61533011, U1806203

This paper investigates the delay-distribution-dependent exponential synchronization problem for a class of chaotic neural networks with mixed random time-varying delays as well as restricted disturbances. Given the probability distribution of the time-varying delay, stochastic variable that satisfying Bernoulli distribution is formulated to produce a new system which includes the information of the probability distribution. Based on the Lyapunov-Krasovskii functional method, the Jensen's integral inequality theory and linear matrix inequality (LMI) technique, several delay-distribution-dependent sufficient conditions are developed to guarantee that the chaotic neural networks with mixed random time-varying delays are exponentially synchronized in mean square. Furthermore, the derived results are given in terms of simplified LMI, which can be straightforwardly solved by Matlab. Finally, two numerical examples are proposed to demonstrate the feasibility and the effectiveness of the presented synchronization scheme.

Citation: 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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020221
##### References:
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show all references

##### References:
 [1] P. Balasubramaniam, V. Vembarasan and R. Rakkiyappan, Delay-dependent robust exponential state estimation of Markovian jumping fuzzy Hopfield neural networks with mixed random time-varying delays, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 2109-2129.  doi: 10.1016/j.cnsns.2010.08.024.  Google Scholar [2] H. Bao and J. Cao, Delay-distibution-dependent state estimation for discrete-time stochastic neural networks with random delay, Neural Network, 24 (2011), 19-28.   Google Scholar [3] J. Cai, L. Shen and F. Wu, Adaptive control of a class of non-linear systems preceded by backlash-like hysteresis, Math. Struct. Comput. Sci., 24 (2014), e240504, 14 pp. doi: 10.1017/S0960129512000473.  Google Scholar [4] A. Cichoki and R. Unbehauen, Neural Networks for Optimization and Signal Processing, John Wiley and Sons, 2003. Google Scholar [5] Q. Gan and Y. Liang, Synchronization of chaotic neural networks with time delay in the leakage term and parametric uncertainties based on sampled-data control, J. Franklin Inst., 349 (2012), 1955-1971.  doi: 10.1016/j.jfranklin.2012.05.001.  Google Scholar [6] M. Gilli, Strange attractors in delayed cellular neural networks, IEE Trans. Circuits Syst. I: Fundam. Theor. Appl., 40 (1993), 849-853.  doi: 10.1109/81.251826.  Google Scholar [7] K. Gu, V. L. Kharitonov and J. Chen, Stability of Time-Delay Systems, Birkhäuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-1-4612-0039-0.  Google Scholar [8] B. Hu, Q. Song, K. Li, Z. Zhao, Y. Liu and Fuad E. Alsaadi, Global $\mu$-synchronization of impulsive complex-valued neural networks with leakage delay and mixed time-varying delays, Neurocomputing, 307 (2018), 106-116.   Google Scholar [9] T. Kwork and K. A. Smith, A unified framework for chaotic neural networks approaches to combinatorial optimization, IEE Trans. Neural Netw., 10 (1999), 978-981.   Google Scholar [10] X. Li and S. Song, Research on synchronization of chaotic delayed neural networks with stochastic perturbation using impulsive control method, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 3892-3900.  doi: 10.1016/j.cnsns.2013.12.012.  Google Scholar [11] J.-N. Li, Y.-J. Su and C.-L. Wen, Stochastic reliable control of a class of networked control systems with actuator faults and input saturation, Int. J. Control, Autom., Syst., 12 (2014), 564-571.  doi: 10.1007/s12555-013-0371-7.  Google Scholar [12] Z. X. Liu, H. W. Yang and F. W. Chen, Mean square exponential synchronization of stochastic neutral type chaotic neural networks with mixed delay, International Journal of Mathematical and Computational Sciences, 8 (2011), 1298-1303.   Google Scholar [13] H. Lu, Chaotic attractors in delayed neural networks, Phys. Lett. A, 298 (2002), 109-116.  doi: 10.1016/S0375-9601(02)00538-8.  Google Scholar [14] G. Nagamani and S. Ramasamy, Stochastic deissativity and passivity analysis for discrete-time neural networks with probabilistic time-varying delays in leakage term, Appl. Math. Comput., 289 (2016), 237-257.  doi: 10.1016/j.amc.2016.05.004.  Google Scholar [15] J. Nilsson, B. Bernhardsson and B. Wittenmark, Stochastic analysis and control of real-time systems with random time delays, Automatica J. IFAC, 34 (1998), 57-64.  doi: 10.1016/S0005-1098(97)00170-2.  Google Scholar [16] C. Peng and Y.-C. Tian, Improved delay-dependent robust stability criteria for uncertain systems with interval time-varing delay, IET Control Theory Appl., 2 (2008), 752-761.  doi: 10.1049/iet-cta:20070362.  Google Scholar [17] A. Pratap, R. Raja, Ji nde Cao, G. Rajchakit and Fuad E. Alsaadi, Further synchronization in finite time analysis for time-varying delayed fractional order memristive competitive neural networks with leakage delay, Neurocomputing, 317 (2018), 110-126.  doi: 10.1016/j.neucom.2018.08.016.  Google Scholar [18] Y. Tang, J. Fang and Q. Miao, On the exponential synchronization of stochastic jumping chaotic neural networks with mixed delays and sector-bounded nonlinearities, Neurocomputing, 721 (2009), 694-701.   Google Scholar [19] G. Velmurugan, R. Rakkiyappan and J. Cao, Finite-time synchronization of fractional-order memristor-based neural networks with time delays, Neural Netw., 73 (2016), 36-46.   Google Scholar [20] J. Wang, K. Shi, Q. Huang, S. Zhong and D. Zhang, Stochastic switched sampled-data control for synchronization of delayed chaotic neural networks with packet dropout, Appl. Math. Comput., 335 (2018), 211-230.  doi: 10.1016/j.amc.2018.04.038.  Google Scholar [21] Z. Wang, H. Shu, Y. Liu, D. W. C. Ho and X. Liu, Robust stability analysis of generalized neural networks with dicrete and distributed time delays, Chaos Solitons Fractals, 30 (2006), 886-896.  doi: 10.1016/j.chaos.2005.08.166.  Google Scholar [22] W. Wang, M. Yu, X. Luo, L. Liu, M. Yuan and W. Zhao, Synchronization of memristive BAM neural networks with leakage delay and additive time-varying delay components via sampled-data control, Chaos Solitons Fractals, 104 (2017), 84-97.  doi: 10.1016/j.chaos.2017.08.011.  Google Scholar [23] H. Wu, X. Zhang, R. Li and R. Yao, Finite-time synchronization of chaotic neural networks with mixed time-varying delays and stochastic disturbance, Memetic Comp., 7 (2015), 231-240.  doi: 10.1007/s12293-014-0150-x.  Google Scholar [24] S. Xu and T. Chen, Robust $H_{\infty}$ control for uncertain stochastic systems with state delay, IEEE Trans. Automat. Control, 47 (2002), 2089-2094.  doi: 10.1109/TAC.2002.805670.  Google Scholar [25] X. Zhang, X. Lv and X. Li, Sampled-data-based lag synchronization of chaotic delayed neural networks with impulsive control, Nonlinear Dyn., 90 (2017), 2199-2207.  doi: 10.1007/s11071-017-3795-4.  Google Scholar [26] C.-D. Zheng, Z. Wei and Z. Wang, Robustly adaptive synchronization for stochastic Markovian neural networks of neutral type with mixed mode-dependent delays, Neurocomputing, 171 (2016), 1254-1264.  doi: 10.1016/j.neucom.2015.07.066.  Google Scholar [27] F. Zou and J. A. Nossek, Bifurcation and chaos in cellular neural networks, IEE Trans. Circuits Syst. I: Fundam. Theor. Appl., 40 (1993), 166-173.  doi: 10.1109/81.222797.  Google Scholar
(a) Chaotic behavior of the neural networks (3). (b) Chaotic behavior of the neural networks (4) without control input $u(t)$
(a) Chaotic behavior of the neural networks (4). (b)-(c) State trajectories of the neural networks (3) and (4). (d) Synchronization error trajectories of the state variables between the neural networks (3) and (4)
(a) Chaotic behavior of the neural networks (3). (b) Chaotic behavior of the neural networks (4) without control input $u(t)$
(a) Chaotic behavior of the neural networks (4). (b)-(c) State trajectories of the neural networks (3) and (4). (d) Synchronization error trajectories of the state variables between the neural networks (3) and (4)
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