Synchronization in an array of coupled neural networks with delayed impulses: Average impulsive delay method

In the paper, synchronization of coupled neural networks with delayed impulses is investigated. In order to overcome the difficulty that time delays can be flexible and even larger than impulsive interval, we propose a new method of average impulsive delay (AID). By the methods of average impulsive interval (AII) and AID, some sufficient synchronization criteria for coupled neural networks with delayed impulses are obtained. We prove that the time delay in impulses can play double roles, namely, it may desynchronize a synchronous network or synchronize a nonsynchronized network. Moreover, a unified relationship is established among AII, AID and rate coefficients of the impulsive dynamical network such that the network is globally exponentially synchronized (GES). Further, we discuss the case that time delays in impulses may be unbounded, which has not been considered in existing results. Finally, two examples are presented to demonstrate the validity of the derived results.

[1]  Xinzhi Liu,et al.  Delay-Dependent Impulsive Distributed Synchronization of Stochastic Complex Dynamical Networks With Time-Varying Delays , 2019, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[2]  Dragan Nesic,et al.  Input-output stability properties of networked control systems , 2004, IEEE Transactions on Automatic Control.

[3]  Fuad E. Alsaadi,et al.  Finite-Time Synchronization of Networks via Quantized Intermittent Pinning Control , 2018, IEEE Transactions on Cybernetics.

[4]  Jianlong Qiu,et al.  Exponential synchronization of time-varying delayed complex-valued neural networks under hybrid impulsive controllers , 2019, Neural Networks.

[5]  David J. Hill,et al.  Event‐triggered control via impulses for exponential stabilization of discrete‐time delayed systems and networks , 2019, International Journal of Robust and Nonlinear Control.

[6]  Jianquan Lu,et al.  Output Tracking of Boolean Control Networks Driven by Constant Reference Signal , 2019, IEEE Access.

[7]  Jinde Cao,et al.  Synchronization of Coupled Markovian Reaction–Diffusion Neural Networks With Proportional Delays Via Quantized Control , 2019, IEEE Transactions on Neural Networks and Learning Systems.

[8]  Vladimir L. Kharitonov,et al.  Stability of Time-Delay Systems , 2003, Control Engineering.

[9]  Jinde Cao,et al.  An Impulsive Delay Inequality Involving Unbounded Time-Varying Delay and Applications , 2017, IEEE Transactions on Automatic Control.

[10]  Chuandong Li,et al.  Synchronization of coupled memristive chaotic circuits via state-dependent impulsive control , 2017 .

[11]  João Pedro Hespanha,et al.  Lyapunov conditions for input-to-state stability of impulsive systems , 2008, Autom..

[12]  Bin Liu,et al.  Stabilisation to input‐to‐state stability for continuous‐time dynamical systems via event‐triggered impulsive control with three levels of events , 2018, IET Control Theory & Applications.

[13]  Jinde Cao,et al.  Synchronization Control for Nonlinear Stochastic Dynamical Networks: Pinning Impulsive Strategy , 2012, IEEE Transactions on Neural Networks and Learning Systems.

[14]  Fuad E. Alsaadi,et al.  Unified synchronization criteria in an array of coupled neural networks with hybrid impulses , 2018, Neural Networks.

[15]  Fuad E. Alsaadi,et al.  Synchronization of dynamical networks with nonlinearly coupling function under hybrid pinning impulsive controllers , 2018, J. Frankl. Inst..

[16]  Xinzhi Liu,et al.  Robust impulsive synchronization of uncertain dynamical networks , 2005, IEEE Transactions on Circuits and Systems I: Regular Papers.

[17]  Shouming Zhong,et al.  Synchronization of nonlinear complex dynamical systems via delayed impulsive distributed control , 2018, Appl. Math. Comput..

[18]  Quanxin Zhu,et al.  Stabilization of stochastic functional differential systems with delayed impulses , 2019, Appl. Math. Comput..

[19]  Jinde Cao,et al.  Pth Moment Exponential Stochastic Synchronization of Coupled Memristor-based Neural Networks with Mixed Delays via Delayed Impulsive Control , 2015, Neural Networks.

[20]  Jinde Cao,et al.  Pinning Synchronization of Nonlinear Coupled Lur’e Networks Under Hybrid Impulses , 2019, IEEE Transactions on Circuits and Systems II: Express Briefs.

[21]  M. Newman,et al.  Scaling and percolation in the small-world network model. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[22]  Wei Xing Zheng,et al.  Input-to-state stability and integral input-to-state stability of nonlinear impulsive systems with delays , 2009, Autom..

[23]  Jurgen Kurths,et al.  Synchronization in complex networks , 2008, 0805.2976.

[24]  Xiaodi Li,et al.  Stabilization of Delay Systems: Delay-Dependent Impulsive Control , 2017, IEEE Transactions on Automatic Control.

[25]  Dan Wei,et al.  Global exponential synchronization of nonlinear time-delay Lur'e systems via delayed impulsive control , 2014, Commun. Nonlinear Sci. Numer. Simul..

[26]  Yuanyuan Li,et al.  Impulsive Synchronization of Stochastic Neural Networks via Controlling Partial States , 2017, Neural Processing Letters.

[27]  Jinde Cao,et al.  pth moment exponential synchronization for stochastic delayed Cohen–Grossberg neural networks with Markovian switching , 2011, Nonlinear Dynamics.

[28]  Quanxin Zhu,et al.  Exponential synchronization of Markovian jumping chaotic neural networks with sampled-data and saturating actuators , 2017 .

[29]  Jinde Cao,et al.  Robust fixed-time synchronization for uncertain complex-valued neural networks with discontinuous activation functions , 2017, Neural Networks.

[30]  Xiaodi Li,et al.  Exponential Stability of Nonlinear Systems With Delayed Impulses and Applications , 2019, IEEE Transactions on Automatic Control.

[31]  Zengrong Liu,et al.  Exponential cluster synchronization of hybrid-coupled impulsive delayed dynamical networks: average impulsive interval approach , 2016 .

[32]  Peng Li,et al.  Input/output-to-state stability of impulsive switched systems , 2018, Syst. Control. Lett..

[33]  Jinde Cao,et al.  A unified synchronization criterion for impulsive dynamical networks , 2010, Autom..

[34]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[35]  Fuad E. Alsaadi,et al.  Adaptive Neural State-Feedback Tracking Control of Stochastic Nonlinear Switched Systems: An Average Dwell-Time Method , 2019, IEEE Transactions on Neural Networks and Learning Systems.

[36]  Kexue Zhang,et al.  Synchronization of linear dynamical networks on time scales: Pinning control via delayed impulses , 2016, Autom..

[37]  Zhenkun Huang,et al.  Scale-Limited Activating Sets and Multiperiodicity for Threshold-Linear Networks on Time Scales , 2014, IEEE Transactions on Cybernetics.

[38]  Xinsong Yang,et al.  Synchronization of uncertain hybrid switching and impulsive complex networks , 2018, Applied Mathematical Modelling.

[39]  Zhigang Zeng,et al.  Stabilization of Second-Order Memristive Neural Networks With Mixed Time Delays via Nonreduced Order , 2020, IEEE Transactions on Neural Networks and Learning Systems.

[40]  Tianping Chen,et al.  New approach to synchronization analysis of linearly coupled ordinary differential systems , 2006 .

[41]  Quanxin Zhu,et al.  pth Moment exponential stability of impulsive stochastic functional differential equations with Markovian switching , 2014, J. Frankl. Inst..

[42]  Jinde Cao,et al.  Synchronizing Neural Networks With Proportional Delays Based on a Class of $q$-Type Allowable Time Scales , 2018, IEEE Transactions on Neural Networks and Learning Systems.

[43]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.

[44]  M. Benchohra,et al.  Impulsive differential equations and inclusions , 2006 .

[45]  Quanxin Zhu,et al.  Some Improved Razumikhin Stability Criteria for Impulsive Stochastic Delay Differential Systems , 2019, IEEE Transactions on Automatic Control.

[46]  Xiaodi Li,et al.  Impulsive control of unstable neural networks with unbounded time-varying delays , 2017, Science China Information Sciences.

[47]  Jinde Cao,et al.  Exponential Synchronization of Linearly Coupled Neural Networks With Impulsive Disturbances , 2011, IEEE Transactions on Neural Networks.

[48]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[49]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.