Science China Information Sciences

, Volume 53, Issue 4, pp 813–822 | Cite as

Pinning control of general complex dynamical networks with optimization

Research Papers


This paper addresses optimal pinning control of general complex dynamical networks. A pinning scheme with linear feedback is proposed to globally exponentially stabilize a network onto a homogeneous state. In particular, we answer an important and fundamental question about pinning control: how to select an optimal combination between the number of pinned nodes and the feedback control gain? Three illustrative examples are provided to show the effectiveness of the proposed technique.


complex dynamical networks optimization pinning control linear feedback 


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  1. 1.
    Barrat A, Weigt M. On the properties of small world networks. Eur Phys J B, 2000, 13: 547–560CrossRefGoogle Scholar
  2. 2.
    Strogatz S H. Exploring complex networks. Nature, 2001, 410: 268–276CrossRefGoogle Scholar
  3. 3.
    Watts D J, Strogatz S H. Collective dynamics of ’small-world’ networks. Nature, 1998, 393: 440–442CrossRefGoogle Scholar
  4. 4.
    Barabási A L, Albert R. Emergence of scaling in random networks. Science, 1999, 286: 509–512CrossRefMathSciNetGoogle Scholar
  5. 5.
    Pecora L M, Carroll T L. Master stability function for synchronized coupled systems. Phys Rev Lett, 1998, 80: 2109–2112CrossRefGoogle Scholar
  6. 6.
    Wu C W, Chua L O. Synchronization in an array of linearly coupled dynamical systems. IEEE Trans Circ Syst-I, 1995, 42: 430–447MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Belykh V N, Belykh I V, Hasler M. Connection graph stability method for synchronized coupled chaotic systems. Physica D, 2004, 195: 159–187MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Pastor-Satorras R, Vespignani A. Epidemic spread in scale-free networks. Phys Rev Lett, 2001, 86: 3200–3203CrossRefGoogle Scholar
  9. 9.
    Wang X F, Chen G. Synchronization in scale-free dynamical networks: robustness and fragility. IEEE Trans Circ Syst-I, 2002, 49: 54–62CrossRefGoogle Scholar
  10. 10.
    Zhou J, Chen T. Synchronization in general complex delayed dynamical networks. IEEE Trans Circ Syst-I, 2006, 53: 733–744CrossRefMathSciNetGoogle Scholar
  11. 11.
    Li Z, Chen G. Global synchronization and asymptotic stability of complex dynamical networks. IEEE Trans Circ Syst-II, Exp Briefs, 2006, 53: 28–33CrossRefGoogle Scholar
  12. 12.
    Wang W, Slotine J J E. Contraction analysis of time-delayed communications and group cooperation. IEEE Trans Autom Control, 2006, 51: 712–717CrossRefMathSciNetGoogle Scholar
  13. 13.
    Lü J, Yu X, Chen G. Chaos synchronization of general complex dynamical networks. Physica A, 2004, 33: 281–302CrossRefGoogle Scholar
  14. 14.
    Lü J, Chen G. A time-varying complex dynamical network models and its controlled synchronization criteria. IEEE Trans Autom Control, 2005, 50: 841–846CrossRefGoogle Scholar
  15. 15.
    Han X P, Lu J A. The changes on synchronizing ability of coupled networks from ring networks to chain networks. Sci China Ser F-Inf Sci, 2007, 50: 615–624MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Wu J, Jiao L. Synchronization in complex dynamical networks with nonsymmetric coupling. Physica D, 2008, 237: 2487–2498MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Xiang L, Liu Z, Chen Z, et al. Pinning weighted complex networks with heterogeneous delays by a small number of feedback controllers. Sci China Ser F-Inf Sci, 2008, 51: 511–523MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Zhan M, Gao J, Wu Y, et al. Chaos synchronization in coupled systems by applying pinning control. Phys Rev E, 2007, 76: 036203CrossRefGoogle Scholar
  19. 19.
    Wang L, Dai H P, Dong H, et al. Adaptive synchronization of weighted complex dynamical networks through pinning. Eur Phys J B, 2008, 61: 335–342CrossRefGoogle Scholar
  20. 20.
    Wang X F, Chen G. Pinning control of scale-free dynamical networks. Physica A, 2002, 310: 521–531MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Li X, Wang X F, Chen G. Pinning a complex dynamical network to its equilibrium. IEEE Trans Circ Syst-I, 2004, 51: 2074–2087CrossRefMathSciNetGoogle Scholar
  22. 22.
    Chen T, Liu X, Lu W. Pinning complex networks by a single controller. IEEE Trans Circ Syst-I, 2007, 54: 1317–1326CrossRefMathSciNetGoogle Scholar
  23. 23.
    Zhou J, Lu J A, Lü J. Pinning adaptive synchronization of a general complex dynamical network. Automatica, 2008, 44: 996–1003Google Scholar
  24. 24.
    Zhou J, Lu J A, Lü J. Erratum to: Pinning adaptive synchronization of a general complex dynamical network. Automatica, 2009, 45: 598–599MATHCrossRefGoogle Scholar
  25. 25.
    Zhou J, Wu X Q, Yu W W, et al. Pinning synchronization of delayed neural networks. Chaos, 2008, 18: 043111CrossRefMathSciNetGoogle Scholar
  26. 26.
    Zhao J C, Lu J A, Zhang Q J. Pinning a complex delayed dynamical network to a homogenous trajectory. IEEE Trans Circ Syst-II: Exp Briefs, 2009, 56: 514–517CrossRefGoogle Scholar
  27. 27.
    Tang Y, Wang Z, Fang J. Pinning control of fractional-order weighted complex networks. Chaos, 2009, 19: 013112CrossRefMathSciNetGoogle Scholar
  28. 28.
    Xia W, Cao J. Pinning synchronization of delayed dynamical networks via periodically intermittent control. Chaos, 2009, 19: 013120CrossRefMathSciNetGoogle Scholar
  29. 29.
    Guo W, Austin F, Chen S, et al. Pinning synchronization of the complex networks with non-delayed and delayed coupling. Phys Lett A, 2009, 373: 1565–1572CrossRefMathSciNetGoogle Scholar
  30. 30.
    Song Q, Cao J. On pinning synchronization of directed and undirected complex dynamical networks. IEEE Trans Circ Syst-I, 2009, in pressGoogle Scholar
  31. 31.
    Lu W, Chen T, Chen G. Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay. Physica D, 2006, 221: 118–134MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Khalil H. Nonlinear systems. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 2002MATHGoogle Scholar
  33. 33.
    Zou F, Nossek J A. Bifurcation and chaos in cellular neural networks. IEEE Trans Circ Syst-I, 1993, 40: 166–173MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Wilkinson J H. The Algebraic Eigenvalue Problem. Oxford: Oxford University, 1965MATHGoogle Scholar

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© Science in China Press and Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina

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