Full Text:   <2809>

CLC number: TP13

On-line Access: 2011-06-07

Received: 2010-05-28

Revision Accepted: 2010-10-18

Crosschecked: 2011-05-05

Cited: 1

Clicked: 6613

Citations:  Bibtex RefMan EndNote GB/T7714

-   Go to

Article info.
1. Reference List
Open peer comments

Journal of Zhejiang University SCIENCE C 2011 Vol.12 No.6 P.470-477


Number estimation of controllers for pinning a complex dynamical network

Author(s):  Lei Wang, Huan Shi, You-xian Sun

Affiliation(s):  School of Mathematics and Systems Science and LMIB, Beihang University, Beijing 100191, China, State Key Lab of Industrial Control Technology, Institute of Industrial Process Control, Zhejiang University, Hangzhou 310027, China

Corresponding email(s):   lwang@buaa.edu.cn

Key Words:  Pinning control, Number estimation, Synchronization, Complex networks

Lei Wang, Huan Shi, You-xian Sun. Number estimation of controllers for pinning a complex dynamical network[J]. Journal of Zhejiang University Science C, 2011, 12(6): 470-477.

@article{title="Number estimation of controllers for pinning a complex dynamical network",
author="Lei Wang, Huan Shi, You-xian Sun",
journal="Journal of Zhejiang University Science C",
publisher="Zhejiang University Press & Springer",

%0 Journal Article
%T Number estimation of controllers for pinning a complex dynamical network
%A Lei Wang
%A Huan Shi
%A You-xian Sun
%J Journal of Zhejiang University SCIENCE C
%V 12
%N 6
%P 470-477
%@ 1869-1951
%D 2011
%I Zhejiang University Press & Springer
%DOI 10.1631/jzus.C1010247

T1 - Number estimation of controllers for pinning a complex dynamical network
A1 - Lei Wang
A1 - Huan Shi
A1 - You-xian Sun
J0 - Journal of Zhejiang University Science C
VL - 12
IS - 6
SP - 470
EP - 477
%@ 1869-1951
Y1 - 2011
PB - Zhejiang University Press & Springer
ER -
DOI - 10.1631/jzus.C1010247

number estimation of controllers is a fundamental question in pinning synchronization of complex networks. This paper studies the problem of controller number in synchronizing a complex network of coupled dynamical systems by means of pinning. For a complex network with a symmetric coupling matrix and full coupling between the nodes, we formulate network synchronization via pinning as a linear matrix inequality criterion, and provide a lower bound and an upper bound of the controller number for a given complex network with fixed architecture. Several numerical examples with Barabási-Albert network topologies are provided to verify our theoretical results.

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article


[1]Albert, R., Barabási, A.L., 2002. Statistical mechanics of complex networks. Rev. Modern Phys., 74(1):47-97.

[2]Arenas, A., Diaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C., 2008. Synchronization in complex networks. Phys. Rep., 469(3):95-153.

[3]Barabási, A.L., Albert, R., 1999. Emergence of scaling in random networks. Science, 286(5439):509-512.

[4]Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., Hwang, D.U., 2006. Complex networks: structure and dynamics. Phys. Rep., 424(4-5):178-308.

[5]Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V., 1994. Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia, USA.

[6]Chen, M.Y., Zhou, D.H., 2006. Synchronization in uncertain complex networks. Chaos, 16(1):013101.

[7]Chen, T., Liu, X., Lu, W., 2007. Pinning complex networks by a single controller. IEEE Trans. Circ. Syst. I, 54(6):1317-1326.

[8]Duan, Z., Wang, J., Chen, G., Huang, L., 2008. Stability analysis and decentralized control of a class of complex dynamical networks. Automatica, 44(4):1028-1035.

[9]Grigoriev, R.O., Cross, M.C., Schuster, H.G., 1997. Pinning control of spatiotemporal chaos. Phys. Rev. Lett., 79(15):2795-2798.

[10]Hu, G., Yang, J., Liu, W., 1998. Instability and controllability of linearly coupled oscillators: eigenvalue analysis. Phys. Rev. E, 58(4):4440-4453.

[11]Kurths, J., Maraun, D., Zhou, C.S., Zamora-Lopez, G., Zou, Y., 2009. S dynamics in complex systems. Eur. Rev., 17(2):357-370.

[12]Li, D., Lu, J., Wu, X., Chen, G., 2006. Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system. J. Math. Anal. Appl., 323(2):844-853.

[13]Li, X., Wang, X., Chen, G., 2004. Pinning a complex dynamical network to its equilibrium. IEEE Trans. Circ. Syst. I, 51(10):2074-2087.

[14]Li, Z., Chen, G., 2004. Robust adaptive synchronization of uncertain dynamical networks. Phys. Lett. A, 324(2-3):166-178.

[15]Lü, J., Chen, G., 2002. A new chaotic attractor coined. Int. J. Bifurc. Chaos, 12(3):659-661.

[16]Lü, J., Chen, G., 2005. A time-varying complex dynamical network model and its controlled synchronization criteria. IEEE Trans. Automat. Control, 50(6):841-846.

[17]Newman, M.E.J., 2003. The structure and function of complex networks. SIAM Rev., 45(2):167-256.

[18]Parekh, N., Parthasarathy, S., Sinha, S., 1998. S global and local control of spatiotemporal chaos in coupled map lattices. Phys. Rev. Lett., 81(7):1401-1404.

[19]Pecora, L.M., Carroll, T.L., 1998. Master stability functions for synchronized coupled systems. Phys. Rev. Lett., 80(10):2109-2112.

[20]Pikovsky, A., Rosenblum, M., Kurths, J., 2001. Synchronization, a Universal Concept in Nonlinear Sciences. Cambridge University Press, Cambridge, UK.

[21]Sorrentino, F., di Benardo, M., Garofalo, F., Chen, G., 2007. Controllability of complex networks via pinning. Phys. Rev. E, 75(4):046103.

[22]Wang, L., Dai, H.P., Dong, H., Cao, Y.Y., Sun, Y.X., 2008a. Adaptive synchronization of weighted complex dynamical networks through pinning. Eur. Phys. J. B, 61(3):335-342.

[23]Wang, L., Kong, X.J., Shi, H., Dai, H.P., Sun, Y.X., 2008b. LMI-based criteria for synchronization of complex dynamical networks. J. Phys. A: Math. Theor., 41(28):285102.

[24]Wang, L., Dai, H.P., Kong, X.J., Sun, Y.X., 2009. Synchronization of uncertain complex dynamical networks via adaptive control. Int. J. Robust Nonlinear Control, 19(5):495-511.

[25]Wang, X., Chen, G., 2002. Pinning control of scale-free dynamical networks. Phys. A, 310(3-4):521-531.

[26]Wu, C.W., 2008. On the relationship between pinning control effectiveness and graph topology in complex networks of dynamical systems. Chaos, 18(3):037103.

[27]Wu, C.W., Chua, L.O., 1995. Synchronization in an array of linearly coupled dynamical systems. IEEE Trans. Circ. Syst. I, 42(8):430-447.

[28]Xiang, J., Chen, G., 2007. On the V-stability of complex dynamical networks. Automatica, 43(6):1049-1057.

[29]Xiang, L.Y., Liu, Z.X., Chen, Z.Q., Chen, F., Yuan, Z.Z., 2007. Pinning control of complex dynamical networks with general topology. Phys. A, 379(1):298-306.

[30]Zhou, J., Lu, J., Lü, J., 2006. Adaptive synchronization of an uncertain complex dynamical network. IEEE Trans. Automat. Control, 51(4):652-656.

[31]Zhou, J., Lu, J., Lü, J., 2008. Pinning adaptive synchronization of a general complex dynamical network. Automatica, 44(4):996-1003.

Open peer comments: Debate/Discuss/Question/Opinion


Please provide your name, email address and a comment

Journal of Zhejiang University-SCIENCE, 38 Zheda Road, Hangzhou 310027, China
Tel: +86-571-87952783; E-mail: cjzhang@zju.edu.cn
Copyright © 2000 - 2024 Journal of Zhejiang University-SCIENCE