Finite-Time \({H_\infty }\) Synchronization for Complex Dynamical Networks with Markovian Jump Parameter

  • Nannan MaEmail author
  • Zhibin Liu
  • Lin Chen


In this paper, the problem of finite-time \({H_\infty }\) synchronization for complex dynamical networks with Markovian jump parameter is investigated. This purpose is concentrated on designing controller such that the obtain synchronization error system is finite-time \({H_\infty }\) synchronization. Based on the delay subinterval decomposition approach and linear matrix inequality approach, a new Lyapunov–Krasovskii functional is proposed to acquire the sufficient condition. Finally, numerical simulations are exploited to demonstrate our theoretical results.


Synchronization Complex dynamical networks Delay subinterval decomposition Linear matrix inequality 



This paper is supported by applied fundamental research (Major frontier projects) of Sichuan Province (16JC0314). The authors would like to thank the editor and anonymous reviewers for their many helpful comments and suggestions to improve the quality of this paper.


  1. Ali, M. S., Saravanakumar, R., & Zhu, Q. X. (2015). Less conservation delay-dependent control of uncertain neural networks with discrete interval and distributed time-varying delays. Neurocomputing, 166, 84–95.CrossRefGoogle Scholar
  2. Balasubramaniam, P., & Chandran, R. (2011). Delay decomposition approach to stability analysis for uncertain fuzzy Hopfield neural networks with time-varying delay. Communications in Nonlinear Science and Numerical Simulation, 16, 2098–2108.MathSciNetCrossRefzbMATHGoogle Scholar
  3. Chen, C., Li, L. X., Peng, H. P., et al. (2017). Finite time synchronization of memristor-based Cohen-Grossberg neural networks with mixed delays. PLoS ONE, 12(9), e0185007.CrossRefGoogle Scholar
  4. Chen, W. H., Jiang, Z. Y., Lu, X. M., & Luo, S. X. (2015). Synchronization for complex dynamical networks with coupling delays using distributed impulsive control. Nonlinear Analysis: Hybrid Systems, 17, 111–127.MathSciNetzbMATHGoogle Scholar
  5. Chen, Y. G., Bi, W. P., & Li, W. L. (2010). Stability analysis for neural networks with time-varying delay: A more general decomposition approach. Neurocomputing, 73, 853–857.CrossRefGoogle Scholar
  6. Cheng, J., Zhu, H., Zhong, S. M., Zeng, Y., & Dong, X. C. (2013). Finite-time control for a class of Markovian jump systems with mode-dependent time-varying delays via new Lyapunov functional. ISA Transactions, 52, 768–774.CrossRefGoogle Scholar
  7. Cheng, M. F., & Hu, H. P. (2011). Synchronization of impulsively-coupled complex switched networks. In Chinese control and decision conference (pp. 177–184).Google Scholar
  8. Cui, W. X., Sun, S. Y., Fang, J. A., Xu, Y. L., & Zhao, L. D. (2014). Finite-time synchronization of Markovian jump complex networks with partially unknown transition rates. Journal of The Franklin Institute, 351, 2543–2561.MathSciNetCrossRefzbMATHGoogle Scholar
  9. D’Addona, D. M., & Teti, R. (2013). Image data processing via neural networks for tool wear prediction. Science Direct, 12, 252–257.Google Scholar
  10. Duan, W., Cai, C., Zou, Y., & You, J. (2013). Synchronization criteria for singular complex dynamical networks with delayed coupling and non-delayed coupling. Control Theory Applications, 30, 947–955.zbMATHGoogle Scholar
  11. Fei, Z., Gao, H., & Shi, P. (2009). New results on stabilization of Markovian jump systems with time delay. Automatica, 45, 2300–2306.MathSciNetCrossRefzbMATHGoogle Scholar
  12. Jing, T. Y., Chen, F. Q., & Li, Q. H. (2015). Finite-time mixed outer synchronization of complex networks with time-varying delay and unknown parameters. Applied Mathematical Modelling, 39, 7734–7743.MathSciNetCrossRefGoogle Scholar
  13. Kalpana, M., Balasubramaniam, P., & Ratnavelu, K. (2015). Sirect delay decomposition approach to synchronization of chaotic fuzzy cellular neural networks with discrete, unbounded distributed delays and Markovian jumping parameters. Applied Mathematica and Computation, 254, 291–304.CrossRefzbMATHGoogle Scholar
  14. Lakshmanan, S., Mathiyalagan, K., Park, J. H., Sakthivel, R., & Rihan, F. A. (2014). Delay-dependent state estimation of neural networks with mixed time-varying delays. Neurpcomputing, 129, 392–400.CrossRefGoogle Scholar
  15. Li, H. J. (2013). Cluster synchronization and state estimation for complex dynamical networks with mixed time delays. Applied Mathematical Modelling, 37, 7223–7244.MathSciNetCrossRefzbMATHGoogle Scholar
  16. Li, D., & Cao, J. D. (2015). Finite-time synchronization of coupled networks with one single time-varying delay coupling. Neurocomputing, 166, 265–270.CrossRefGoogle Scholar
  17. Li, F., & Shen, H. (2015). Finite-time synchronization control for semi-Markov jump delayed neural networks with randomly occurring uncertainties. Neurpcomputing, 166, 447–454.CrossRefGoogle Scholar
  18. Li, Z. K., Duan, Z. S., & Chen, G. R. (2009). Disturbance rejection and pinning control of linear complex dynamical networks. Chinese Physics B, 18, 5228–5234.CrossRefGoogle Scholar
  19. Liu, P. L. (2013a). A delay decomposition approach to stability analysis of neutral systems with time-varying delay. Applied Mathematical Modelling, 37, 5013–5026.MathSciNetCrossRefzbMATHGoogle Scholar
  20. Liu, P. L. (2013b). State feedback stabilization of time-varying delay uncertain system: A delay decomposition approach. Linear Algebra and Its Applications, 438, 2188–2209.MathSciNetCrossRefzbMATHGoogle Scholar
  21. Liu, P. L. (2015). Delayed decomposition approach to the robust absolute stability of a Lur’e control system with time-varying delay. Applied Mathematical Modeling, 00, 1–13.Google Scholar
  22. Liu, K., & Fridman, E. (2012). Networked-based stabilization via discontinuous Lyapunov functional. International Journal of Robust and Nonlinear Control, 22, 420–436.MathSciNetCrossRefzbMATHGoogle Scholar
  23. Liu, X. H., Yu, X. H., & Xi, H. S. (2015). Finite-time synchronization of neural complex networks with Markovian switching based on pinning controller. Neurocomputing, 153, 148–158.CrossRefGoogle Scholar
  24. Lu, P. L., & Yang, Y. (2012). Synchronization of a class of complex networks. In Chinese control conference (pp. 1136–1141).Google Scholar
  25. Ma, N. N., Liu, Z. B., & Chen, L. (2018). Robust and non-fragile finite time \({H_\infty }\) synchronization control for complex networks with uncertain inner coupling. Computational and Applied Mathematics, 37, 5395–5409.MathSciNetCrossRefzbMATHGoogle Scholar
  26. Mei, J., Jiang, M. H., Xu, W. M., & Wang, B. (2013). Finite-time synchronization control of complex networks with time delay. Communications in Nonlinear Science and Numerical Simulation, 18, 2462–2478.MathSciNetCrossRefzbMATHGoogle Scholar
  27. Revathi, V. M., Balasubramaniam, P., & Ratnavelu, K. (2016). Delay-dependent filtering for complex dynamical networks with time-varying delays in nonlinear function and network couplings. Signal Processing, 118, 122–132.CrossRefGoogle Scholar
  28. Shao, Y. Y., Liu, X. D., Sun, X., & Zhang, Q. L. (2014). A delay decomposition approach to admissibility for discrete-time singular delay systems. Information Sciences, 279, 893–905.MathSciNetCrossRefzbMATHGoogle Scholar
  29. Shen, H., Park, J. H., Wu, Z. G., & Zhang, Z. Q. (2015). Finite-time synchronization for complex networks with semi-Markov jump topology. Communications in Nonlinear Science and Numerical Simulation, 24, 40–51.MathSciNetCrossRefGoogle Scholar
  30. Sun, Y. Z., Li, W., & Zhao, D. H. (2012). Finite-time stochastic outer synchronization between two complex dynamical networks with different topologies. Chao, 22, 023152.MathSciNetCrossRefzbMATHGoogle Scholar
  31. Su, L., & Shen, H. (2015). Mixed/passive synchronization for complex dynamical networks with sampled-data control. Applied Mathematical and Computation, 259, 931–942.MathSciNetCrossRefzbMATHGoogle Scholar
  32. Wang, H., & Xue, A. (2011). New stability criterion for singular time-delay systems and its application to partial element equivalent circuit. Control Theory Applications, 28, 1431–1435.Google Scholar
  33. Wu, H. Q., Zhang, X. W., Li, R. X., & Yao, R. (2015). Finite-time synchronization of chaotic neural networks with mixed time-varying delays and stochastic disturbance. Memetic Computing, 7, 1–10.CrossRefGoogle Scholar
  34. Wu, L., Su, X., Shi, P., & Qiu, J. (2011). A new approach to stability analysis and stabilization of discrete-time T–S fuzzy time-varying delay systems. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 40, 273–286.CrossRefGoogle Scholar
  35. Xu, R. P., Kao, Y. G., & Gao, M. M. (2015). Finite-time synchronization of Markovian jump complex networks with generally uncertain transition rates. Journal of Biological Chemistry, 271, 14271–14279.Google Scholar
  36. Xu, Y. H., Zhou, W. N., Fang, J. A., Xie, C. R., & Tong, D. B. (2016). Finite-time synchronization of the complex dynamical network with non-derivative and derivative coupling. Neurocomputing, 173, 1356–1361.CrossRefGoogle Scholar
  37. Yang, X. S., & Cao, J. D. (2010). Finite-time stochastic synchronization of complex networks. Applied Mathematical Modeling, 34, 3631–3641.MathSciNetCrossRefzbMATHGoogle Scholar
  38. Yang, R., Zhang, Z., & Shi, P. (2010). Exponential stability on stochastic neural networks with discrete interval and distributed. IEEE Transactions on Neural Networks, 21, 169–175.CrossRefGoogle Scholar
  39. Zhang, X. M., & Han, Q. L. (2009). A delay decomposition approach to control of networked control systems. European Journal of Control, 5, 523–533.MathSciNetCrossRefzbMATHGoogle Scholar
  40. Zhang, H. T., Yu, T., Sang, J. P., & Zou, X. W. (2014). Dynamic fluctuation model of complex networks with weight scaling behavior and its application to airport networks. Physica A, 39, 500–599.Google Scholar
  41. Zheng, M. W., Li, L. X., Peng, H. P., et al. (2017). Finite-time projective synchronization of memristor-based delay fractional-order neural networks. Nonlinear Dynamics, 89, 2641–2655.MathSciNetCrossRefzbMATHGoogle Scholar
  42. Zheng, M. W., Li, L. X., Peng, H. P., et al. (2018). Finite-time stability and synchronization of memristor-based fractional-order fuzzy cellular neural networks. Communications in Nonlinear Science and Numerical Simulation, 59, 2462–2478.MathSciNetCrossRefGoogle Scholar
  43. Zhu, J. W., & Yang, G. H. (2016). Robust dynamic output feedback synchronization for complex dynamical networks with disturbances. Neurocomputing, 175, 287–292.CrossRefGoogle Scholar

Copyright information

© Brazilian Society for Automatics--SBA 2018

Authors and Affiliations

  1. 1.School of ScienceSouthwest Petroleum UniversityChengduChina

Personalised recommendations