SCIENCE CHINA Information Sciences, Volume 61 , Issue 5 : 052203(2018) https://doi.org/10.1007/s11432-016-9099-x

Guaranteed cost boundary control for cluster synchronization of complex spatio-temporal dynamical networks with community structure

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  • ReceivedNov 23, 2016
  • AcceptedApr 20, 2017
  • PublishedNov 20, 2017



This work was supported by National Natural Science Foundation of China (Grant Nos. 61573096, 61272530, 61703193), Natural Science Foundation of Jiangsu Province of China (Grant No. BK2012741), Natural Science Foundation of Shandong Province of China (Grant Nos. ZR2017MF022, ZR2015FL0 21), Youth Project of National Education Science Fund in the 13th Five-year Plan (Grant No. EIA160450), “333 Engineering” Foundation of Jiangsu Province of China (Grant No. BRA2015286) and National Priority Research Project NPRP funded by Qatar National Research Fund (Grant No. 9 166-1-031).


[1] Garone E, Gasparri A, Lamonaca F. Clock synchronization protocol for wireless sensor networks with bounded communication delays. Automatica, 2015, 59: 60-72 CrossRef Google Scholar

[2] Li H, Liao X, Huang T. Event-Triggering Sampling Based Leader-Following Consensus in Second-Order Multi-Agent Systems. IEEE Trans Automat Contr, 2015, 60: 1998-2003 CrossRef Google Scholar

[3] Cao J D, Rakkiyappan R, Maheswari K. Exponential H filtering analysis for discrete-time switched neural networks with random delays using sojourn probabilities. Sci China Technol Sci, 2016, 59: 387-402 CrossRef Google Scholar

[4] Tingwen Huang , Chuandong Li , Shukai Duan . Robust exponential stability of uncertain delayed neural networks with stochastic perturbation and impulse effects.. IEEE Trans Neural Netw Learning Syst, 2012, 23: 866-875 CrossRef PubMed Google Scholar

[5] Li H, Huang C, Chen G. Distributed Consensus Optimization in Multiagent Networks With Time-Varying Directed Topologies and Quantized Communication.. IEEE Trans Cybern, 2017, 47: 2044-2057 CrossRef PubMed Google Scholar

[6] Li H, Chen G, Huang T. Event-Triggered Distributed Average Consensus Over Directed Digital Networks With Limited Communication Bandwidth.. IEEE Trans Cybern, 2016, 46: 3098-3110 CrossRef PubMed Google Scholar

[7] Li H, Chen G, Huang T. High-Performance Consensus Control in Networked Systems With Limited Bandwidth Communication and Time-Varying Directed Topologies.. IEEE Trans Neural Netw Learning Syst, 2017, 28: 1043-1054 CrossRef PubMed Google Scholar

[8] Wu X J, Zhu C J, Kan H B. An improved secure communication scheme based passive synchronization of hyperchaotic complex nonlinear system. Appl Math Comput, 2015, 252: 201--214. Google Scholar

[9] Fan D G, Wang Q Y. Synchronization and bursting transition of the coupled hindmarsh-rose systems with asymmetrical time-delays. Sci China Technol Sci, 2017, 60: 1019--1031. Google Scholar

[10] Li X, Rakkiyappan R, Sakthivel N. Non-Fragile Synchronization Control For Markovian Jumping Complex Dynamical Networks With Probabilistic Time-Varying Coupling Delays. Asian J Control, 2015, 17: 1678-1695 CrossRef Google Scholar

[11] Yao C, Zhao Q, Yu J. Complete synchronization induced by disorder in coupled chaotic lattices. Phys Lett A, 2013, 377: 370-377 CrossRef ADS Google Scholar

[12] Mahmoud G M, Mahmoud E E. Complex modified projective synchronization of two chaotic complex nonlinear systems. NOnlinear Dyn, 2013, 73: 2231-2240 CrossRef Google Scholar

[13] Rajagopal K, Vaidyanathan S. Adaptive lag synchronization of a modified rucklidge chaotic system with unknown parameters and its labview implementation. Sensor Transducers, 2016, 200: 37--44. Google Scholar

[14] Ferrari F A S, Viana R L, Lopes S R. Phase synchronization of coupled bursting neurons and the generalized Kuramoto model.. Neural Networks, 2015, 66: 107-118 CrossRef PubMed Google Scholar

[15] Li T, Rao B. Criteria of Kalman's Type to the Approximate Controllability and the Approximate Synchronization for a Coupled System of Wave Equations with Dirichlet Boundary Controls. SIAM J Control Optim, 2016, 54: 49-72 CrossRef Google Scholar

[16] Yu W, Chen G, Lü J. On pinning synchronization of complex dynamical networks. Automatica, 2009, 45: 429-435 CrossRef Google Scholar

[17] Cao J, Li R. Fixed-time synchronization of delayed memristor-based recurrent neural networks. Sci China Inf Sci, 2017, 60: 032201 CrossRef Google Scholar

[18] Nagornov R, Osipov G, Komarov M. Mixed-mode synchronization between two inhibitory neurons with post-inhibitory rebound. Commun NOnlinear Sci Numer Simul, 2016, 36: 175-191 CrossRef ADS Google Scholar

[19] Sorrentino F, Pecora L M, Hagerstrom A M. Complete characterization of the stability of cluster synchronization in complex dynamical networks. Sci Adv, 2016, 2: e1501737-e1501737 CrossRef PubMed ADS arXiv Google Scholar

[20] Wang Y, Cao J. Cluster synchronization in nonlinearly coupled delayed networks of non-identical dynamic systems. NOnlinear Anal-Real World Appl, 2013, 14: 842-851 CrossRef Google Scholar

[21] Kang Y, Qin J, Ma Q. Cluster Synchronization for Interacting Clusters of Nonidentical Nodes via Intermittent Pinning Control.. IEEE Trans Neural Netw Learning Syst, 2018, 29: 1747-1759 CrossRef PubMed Google Scholar

[22] Cao J, Li L. Cluster synchronization in an array of hybrid coupled neural networks with delay.. Neural Networks, 2009, 22: 335-342 CrossRef PubMed Google Scholar

[23] Kaneko K. Relevance of dynamic clustering to biological networks. Physica D-NOnlinear Phenomena, 1994, 75: 55-73 CrossRef ADS Google Scholar

[24] Wang K, Fu X, Li K. Cluster synchronization in community networks with nonidentical nodes. Chaos, 2009, 19: 023106 CrossRef PubMed ADS Google Scholar

[25] Lu W, Liu B, Chen T. Cluster synchronization in networks of coupled nonidentical dynamical systems. Chaos, 2010, 20: 013120 CrossRef PubMed ADS arXiv Google Scholar

[26] Wu Z, Fu X. Cluster mixed synchronization via pinning control and adaptive coupling strength in community networks with nonidentical nodes. Commun NOnlinear Sci Numer Simul, 2012, 17: 1628-1636 CrossRef ADS Google Scholar

[27] Wang J W, Wu H N, Li H X. Guaranteed cost distributed fuzzy observer-based control for a class of nonlinear spatially distributed processes. AIChE J, 2013, 59: 2366-2378 CrossRef Google Scholar

[28] Sheng L, Yang H, Lou X. Adaptive exponential synchronization of delayed neural networks with reaction-diffusion terms. Chaos Solitons Fractals, 2009, 40: 930-939 CrossRef ADS Google Scholar

[29] Yu F, Jiang H. Global exponential synchronization of fuzzy cellular neural networks with delays and reaction-diffusion terms. Neurocomputing, 2011, 74: 509-515 CrossRef Google Scholar

[30] Yang C D, Qiu J L, He H B. Exponential synchronization for a class of complex spatio-temporal networks with space-varying coefficients. Neurocomputing, 2015, 151: 40--47. Google Scholar

[31] Cheng Hu , Haijun Jiang , Zhidong Teng . Impulsive control and synchronization for delayed neural networks with reaction-diffusion terms.. IEEE Trans Neural Netw, 2010, 21: 67-81 CrossRef PubMed Google Scholar

[32] Hu C, Yu J, Jiang H. Exponential synchronization for reaction-diffusion networks with mixed delays in terms of p-norm via intermittent driving.. Neural Networks, 2012, 31: 1-11 CrossRef PubMed Google Scholar

[33] Gan Q. Adaptive synchronization of stochastic neural networks with mixed time delays and reaction-diffusion terms. NOnlinear Dyn, 2012, 69: 2207-2219 CrossRef Google Scholar

[34] Wang J L, Wu H N. Synchronization and adaptive control of an array of linearly coupled reaction-diffusion neural networks with hybrid coupling.. IEEE Trans Cybern, 2014, 44: 1350-1361 CrossRef PubMed Google Scholar

[35] Jin-Liang Wang , Huai-Ning Wu , Lei Guo . Novel adaptive strategies for synchronization of linearly coupled neural networks with reaction-diffusion terms.. IEEE Trans Neural Netw Learning Syst, 2014, 25: 429-440 CrossRef PubMed Google Scholar

[36] Shafi Y, Arcak M. An adaptive algorithm for synchronization in diffusively-coupled systems. In: Proceedings of American Control Conference, Portland, 2014. 2220--2225. Google Scholar

[37] Wu K N, Tian T, Wang L. Synchronization for a class of coupled linear partial differential systems via boundary control. J Franklin Institute, 2016, 353: 4062-4073 CrossRef Google Scholar

[38] Wu K N, Tian T, Wang L. Asymptotical synchronization for a class of coupled time-delay partial differential systems via boundary control. Neurocomputing, 2016, 197: 113-118 CrossRef Google Scholar

[39] Wu Z G, Dong S, Shi P. Fuzzy-Model-Based Nonfragile Guaranteed Cost Control of Nonlinear Markov Jump Systems. IEEE Trans Syst Man Cybern Syst, 2017, 47: 2388-2397 CrossRef Google Scholar

[40] Lu R, Cheng H, Bai J. Fuzzy-Model-Based Quantized Guaranteed Cost Control of Nonlinear Networked Systems. IEEE Trans Fuzzy Syst, 2015, 23: 567-575 CrossRef Google Scholar

[41] Wang J W, Li H X, Wu H N. Fuzzy guaranteed cost sampled-data control of nonlinear systems coupled with a scalar reaction-diffusion process. Fuzzy Sets Syst, 2016, 302: 121-142 CrossRef Google Scholar

[42] Wang Z P, Wu H N. Finite dimensional guaranteed cost sampled-data fuzzy control for a class of nonlinear distributed parameter systems. Inf Sci, 2016, 327: 21-39 CrossRef Google Scholar

[43] Lee T H, Ji D H, Park J H, et al. Decentralized guaranteed cost dynamic control for synchronization of a complex dynamical network with randomly switching topology. Appl Math Comput, 2012, 219: 996--1010. Google Scholar

[44] Feng J W, Yang P, Zhao Y. Cluster synchronization for nonlinearly time-varying delayed coupling complex networks with stochastic perturbation via periodically intermittent pinning control. Appl Math Comput, 2016, 291: 52--68. Google Scholar

[45] Seuret A, Gouaisbaut F. Wirtinger-based integral inequality: Application to time-delay systems. Automatica, 2013, 49: 2860-2866 CrossRef Google Scholar

[46] Yang C, Zhang A, Chen X. Stability and stabilization of a delayed PIDE system via SPID control. Neural Comput Applic, 2017, 28: 4139-4145 CrossRef Google Scholar

[47] Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. Berlin: Springer, 1983. Google Scholar

[48] Zhou J, Lu J, Lu J. Adaptive Synchronization of an Uncertain Complex Dynamical Network. IEEE Trans Automat Contr, 2006, 51: 652-656 CrossRef Google Scholar

[49] Chen Z, Fu X, Zhao D. Anti-periodic mild attractor of delayed hopfield neural networks systems with reaction-diffusion terms. Neurocomputing, 2013, 99: 372-380 CrossRef Google Scholar

[50] Lu J, Ho D W C, Cao J. Single impulsive controller for globally exponential synchronization of dynamical networks. NOnlinear Anal-Real World Appl, 2013, 14: 581-593 CrossRef Google Scholar

[51] Wu H N, Wang J W, Li H X. Fuzzy Boundary Control Design for a Class of Nonlinear Parabolic Distributed Parameter Systems. IEEE Trans Fuzzy Syst, 2014, 22: 642-652 CrossRef Google Scholar

[52] Wang J W, Wu H N, Sun C Y. Boundary controller design and well-posedness analysis of semi-linear parabolic PDE systems. In: Proceedings of 2014 American Control Conference, Beijing, 2014. 3369--3374. Google Scholar

  • Figure 1

    (Color online) The open-loop trajectories of synchronization errors $\left\|~{e_i~(\cdot,~t)}\right\|~$, $i\in~\{1,~2,~\ldots,~6\}$. (a) $\left\|~{e_{i,1}(\cdot,~t)}\right\|,~i\in~V_1$; (b) $\left\|~{e_{i,2}(\cdot,~t)}\right\|,~i\in~V_1$; (c) $\left\|~{e_{i,1}(\cdot,~t)}\right\|,~i\in~V_2$; (d) $\left\|~{e_{i,2}(\cdot,~t)}\right\|,~i\in~V_2$.

  • Figure 2

    (Color online) The closed-loop profiles of synchronization errors $e_i~(X,~t),~i\in~\{1,~2,~\ldots,~6\}$.

  • Figure 3

    (Color online) The trajectories of control inputs $u_i~(t)$, $i\in~\{1,~2,~\ldots,~6\}$.