Control Problems on Time Scales

  • Xinzhi Liu
  • Kexue Zhang
Part of the IFSR International Series in Systems Science and Systems Engineering book series (IFSR, volume 33)


In this chapter, two types of control problems of impulsive systems on time scales are discussed. Section 10.1 formulates the controllability and observability of impulsive time-varying linear systems, and presents the controllability and observability results. In Section 10.2, pinning synchronization of linear dynamical networks (DNs) on time scales is studied. A pinning impulsive control scheme that takes into account time-delay effects is presented to achieve synchronization of DNs on time scales with the state of an isolated node. Based on the theory of time scales and the direct Lyapunov method, a synchronization criterion is established for linear DNs on general time scales. Numerical simulations are given to illustrate the effectiveness of the theoretical analysis.


  1. 14.
    M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications (Birkhäuser, Boston, 2001)CrossRefGoogle Scholar
  2. 37.
    Z. Guan, T. Qian, X. Yu, On controllability and observability for a class of impulsive systems. Syst. Control Lett. 47(3), 247–257 (2002)MathSciNetCrossRefGoogle Scholar
  3. 38.
    Z. Guan, T. Qian, X. Yu, Controllability and observability of linear time varying impulsive systems. IEEE Trans. Circuits Syst. I: Fund. Theory Appl. 49(8), 1198–1208 (2002)MathSciNetCrossRefGoogle Scholar
  4. 47.
    A. Hu, Z. Xu, Pinning a complex dynamical network via impulsive control. Phys. Lett. A 374(2), 186–190 (2009)CrossRefGoogle Scholar
  5. 49.
    M. Hu, L. Wang, Exponential synchronization of chaotic delayed neural networks on time scales. Int. J. Appl. Math. Stat. 34(4),96–103 (2013)MathSciNetGoogle Scholar
  6. 71.
    B. Liu, X. Liu, G. Chen, Robust impulsive synchronization of uncertain dynamical networks. IEEE Trans. Circuits Syst. I: Regul. Pap. 52(7), 1431–1441 (2005)MathSciNetCrossRefGoogle Scholar
  7. 79.
    X. Liu, K. Zhang, Impulsive control for stabilisation of discrete delay systems and synchronisation of discrete delay dynamical networks. IET Control Theory Appl. 8(13), 1185–1195 (2014)MathSciNetCrossRefGoogle Scholar
  8. 91.
    Y. Long, M. Wu, B. Liu, Robust impulsive synchronization of linear discrete dynamical networks. J. Control Theory Appl. 3(1), 20–26 (2005)MathSciNetCrossRefGoogle Scholar
  9. 94.
    J. Lu, D. Ho, J. Cao, J. Kurths, Single impulsive controller for globally exponential synchronization of dynamical networks. Nonlinear Anal. Real World Appl. 14(1), 581–593 (2013)MathSciNetCrossRefGoogle Scholar
  10. 95.
    J. Lu, J. Kurths, J. Cao, N. Mahdavi, C. Huang, Synchronization control for nonlinear stochastic dynamical networks: pinning impulsive strategy. IEEE Trans. Neural Netw. Learn. Syst. 23(2), 285–292 (2012)CrossRefGoogle Scholar
  11. 98.
    T.A. Luk’yanova, A.A. Martynyuk, On the asymptotic stability of a neural network on a time scale. Nonlinear Oscillations 13(3), 372–388 (2010)MathSciNetCrossRefGoogle Scholar
  12. 99.
    V. Lupulescua, A. Younus, Controllability and observability for a class of time-varying impulsive systems on time scales. Electron. J. Qual. Theory Differ. Equ. 95, 1–30 (2011)MathSciNetCrossRefGoogle Scholar
  13. 100.
    V. Lupulescua, A. Younus, Controllability and observability for a class of linear impulsive dynamic systems on time scales. Math. Comput. Model. 54(5–6), 1300–1310 (2011)MathSciNetCrossRefGoogle Scholar
  14. 127.
    Y. Tang, W.K. Wong, J.A. Fang, Q.Y. Miao, Pinning impulsive synchronization of stochastic delayed coupled networks. Chin. Phys. B 20(4), 040513 (10 pp.) (2011)CrossRefGoogle Scholar
  15. 147.
    X. Yang, J. Cao, Z. Yang, Synchronization of coupled reaction-diffusion neural networks with time-varying delays via pinning-impulsive controller. SIAM J. Control Optim. 51(5), 3486–3510 (2013)MathSciNetCrossRefGoogle Scholar
  16. 163.
    Q. Zhang, J. Lu, J. Zhao, Impulsive synchronization of general continuous and discrete-time complex dynamical networks. Commun. Nonlinear Sci. Numer. Simul. 15(4), 1063–1070 (2010)MathSciNetCrossRefGoogle Scholar
  17. 169.
    S. Zhao, J. Sun, Controllability and observability for impulsive systems in complex fields. Nonlinear Anal. Real World Appl. 11(3), 1513–1521 (2010)MathSciNetCrossRefGoogle Scholar
  18. 171.
    J. Zhou, Q. Wu, L. Xiang, Pinning complex delayed dynamical networks by a single impulsive controller. IEEE Trans. Circuits Syst. I Regul. Pap. 58(12), 2882–2893 (2011)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Xinzhi Liu
    • 1
  • Kexue Zhang
    • 1
  1. 1.Department of Applied MathematicsUniversity of WaterlooWaterlooCanada

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