Similar articles

Abstract: 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.

[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.