Instability and Unboundedness Analysis for Impulsive Differential Systems with Applications to Lurie Control Systems

  • Xiaodi LiEmail author
  • Wu-Hua Chen
  • Wei Xing Zheng
  • Qing-guo Wang
Regular Papers Control Theory and Applications


The present paper is dealt with the problem of the instability and unboundedness of impulsive differential systems. Numerous impulsive control systems were reported in the literature. The existing results give sufficient conditions for stability or boundedness of the system. If a system fails such conditions, it might be stable or unstable, bounded or unbounded. The most dangerous among these possible cases are unstable and unbounded ones. To our best knowledge, no criterion is available now to check these two cases. Thus, this paper presents some sufficient conditions on instability and unboundedness of impulsive systems, specializes them to Lurie control systems, and validates them with three illustrative examples.


Comparison method impulsive differential systems (IDSs) instability Lurie control systems unboundedness 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Haykin, Neural Networks, Prentice Hall, New Jersey. NJ, 1999.zbMATHGoogle Scholar
  2. [2]
    B. Liu, C. Dou, and D. Hill, “Robust exponential input-tostate stability of impulsive systems with an application in micro-grids,” Systems & Control Letters, vol. 65, pp. 64–73, 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A. Churilov, A. Medvedev, and P. Mattsson, “Periodical solutions in a pulse modulated model of endocrine regulation with time-delay,” IEEE Transactions on Automatic Control, vol. 59, no. 3, pp. 728–733, 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Z. Liu, S. Zhong, and X. Liu, “Permanence and periodic solutions for an impulsive reaction-diffusion food-chain system with ratio-dependent functional response,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 1, pp. 173–188, 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    X. Li and J. Cao, “An impulsive delay inequality involving unbounded time-varying delay and applications,” IEEE Transactions on Automatic Control, vol. 62, no. 7, pp. 3618–3625, 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    J. Lu, D. Ho, and J. Cao, “A unified synchronization criterion for impulsive dynamical networks,” Automatica vol. 46, no. 7, pp. 1215–1221, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    G. Arthi, Ju H. Park, and H. Y. Jung, “Existence and exponential stability for neutral stochastic integrodifferential equations with impulses driven by a fractional Brownian motion,” Communications in Nonlinear Science and Numerical Simulation, vol. 32, pp. 145–157, 2016.MathSciNetCrossRefGoogle Scholar
  8. [8]
    V. Lakshmikantham, D. Bainov, and P. Simeonov, Series in Modern Applied Mathematics: Vol. 6. Theory of Impulsive Differential Equations, World Scientific Publishing Co, NJ, Teaneck, NJ, 1989.CrossRefGoogle Scholar
  9. [9]
    D. Bainov and P. Simeonov, Systems with Impulse Effect: Stability Theory and Applications, Ellis Horwood, Chichester. UK, 1989.zbMATHGoogle Scholar
  10. [10]
    W. Haddad, V. Chellaboina, and S. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton University Press, New Jersey, NJ, 2006.CrossRefzbMATHGoogle Scholar
  11. [11]
    A. Samoilenko and N. Perestyuk, Impulsive Differential Equations, World Scientific, 1995.CrossRefzbMATHGoogle Scholar
  12. [12]
    T. Yang, Impulsive Control Theory, Springer, Berlin, 2001.zbMATHGoogle Scholar
  13. [13]
    I. Stamova, Stability Analysis of Impulsive Functional Differential Equations, De Gruyter Expositions in Mathematics, Berlin, 2009.CrossRefzbMATHGoogle Scholar
  14. [14]
    X. Liao, L. Wang, and P. Yu, Stability of Dynamical Systems, Elsevier, 2007.CrossRefzbMATHGoogle Scholar
  15. [15]
    W. Zhang, Y. Tang, and X. Wu, “Synchronization of nonlinear dynamical networks with heterogeneous impulses,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 61, no. 4, pp. 1220–1228, 2014.CrossRefGoogle Scholar
  16. [16]
    X. Li, X. Zhang, and S. Song, “Effect of delayed impulses on input-to-state stability of nonlinear systems,” Automatica, vol. 76, pp. 378–382, 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    X. Zhang and X. Li, “Input-to-state stability of non-linear systems with distributed-delayed impulses,” IET Control Theory & Applications, vol. 11, no. 1, pp. 81–89, 2017.MathSciNetCrossRefGoogle Scholar
  18. [18]
    W. Chen, X. Lu, and W. Zheng, “Impulsive stabilization and impulsive synchronization of discrete-time delayed neural networks,” IEEE Transactions on Neural Networks and Learning Systems, vol. 26, no. 4, pp. 734–748, 2015.MathSciNetCrossRefGoogle Scholar
  19. [19]
    F. Amato, R. Ambrosino, M. Ariola, and C. Cosentino, “Finite-time stability of linear time-varying systems with jumps,” Automatica, vol. 45, no. 5, pp. 1354–1358, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Y. Yang, Y. He, Y. Wang, and M. Wu, “Stability analysis for impulsive fractional hybrid systems via variational Lyapunov method,” Communications in Nonlinear Science and Numerical Simulation, vol. 45, pp. 140–157, 2017.MathSciNetCrossRefGoogle Scholar
  21. [21]
    J. Lu, D.W. C. Ho, and J. Cao, “Single impulsive controller for globally exponential synchronization of dynamical networks,” Nonlinear Analysis: Real World Applications, vol. 14, no. 1, pp. 581–593, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Z. Tang, J. H. Park, and J. Feng, “Impulsive effects on quasi-synchronization of neural networks with parameter mismatches and time-varying delay,” IEEE Transactions on Neural Networks and Learning Systems, vol. 29, no. 4, pp. 908–919, 2018.CrossRefGoogle Scholar
  23. [23]
    X. Li and S. Song, “Stabilization of delay systems: delaydependent impulsive control,” IEEE Transactions on Automatic Control, vol. 62, no. 1, pp. 406–411, 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    X. Li and J. Wu, “Stability of nonlinear differential systems with state-dependent delayed impulses,” Automatica, vol. 64, pp. 63–69, Feb 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    F. Yao, J. Cao, P. Cheng, and L. Qiu, “Generalized average dwell time approach to stability and input-to-state stability of hybrid impulsive stochastic differential systems,” Nonlinear Analysis: Hybrid Systems, vol. 22, pp. 147–160, 2016.MathSciNetzbMATHGoogle Scholar
  26. [26]
    W. Chen, S. Luo, and W. X. Zheng, “Generating globally stable periodic solutions of delayed neural networks with periodic coefficients via impulsive control,” IEEE Transactions on Cybernetics, vol. 47, no. 7, pp. 1590–1603, 2017.CrossRefGoogle Scholar
  27. [27]
    X. Li and S. Song, “Impulsive control for existence, uniqueness and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays,” IEEE Transactions on Neural Networks and Learning Systems, vol. 24, no. 6, pp. 868–877, 2013.CrossRefGoogle Scholar
  28. [28]
    V. Lakshmikantham, S. Leela, and S. Kaul, “Comparison principle for impulsive differential equations with variable times and stability theory,” Nonlinear Analysis: Theory, Methods & Applications, vol. 22, no. 4, pp. 499–503, 1994.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    B. Liu and D. J. Hill, “Comparison principle and stability of discrete-time impulsive hybrid systems,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 56, no. 1, pp. 233–245, 2009.MathSciNetCrossRefGoogle Scholar
  30. [30]
    X. Wu, Y. Tang, and W. Zhang, “Input-to-state stability of impulsive stochastic delayed systems under linear assumptions,” Automatica, vol. 66, pp. 195–204, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    B. Zhou and A. Egorov, “Razumikhin and Krasovskii stability theorems for time-varying time-delay systems,” Automatica, vol. 71, pp. 281–291, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    J. Hale and S. Lunel, Introduction to Functional Differential Equations, Springer, Berlin, 1993.CrossRefzbMATHGoogle Scholar
  33. [33]
    X. Chen, J. H. Park, and J. Cao, “Sliding mode synchronization of multiple chaotic systems with uncertainties and disturbances,” Applied Mathematics and Computation, vol. 308, pp. 161–173, 2017.MathSciNetCrossRefGoogle Scholar
  34. [34]
    X. Li, M. Bohner, and C. Wang, “Impulsive differential equations: Periodic solutions and applications.” Automatica, vol. 52, pp. 173–178, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    O. M. Kwon, M. J. Park, and J. H. Park, “New and improved results on stability of static neural networks with interval time-varying delays,” Applied Mathematics and Computation, vol. 239, pp. 346–357, 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, Academic Press, 1969.zbMATHGoogle Scholar
  37. [37]
    Z. Tang, J. H. Park, and H. Shen, “Finite-time cluster synchronization of Lur’e networks: a nonsmooth approach,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, In press.Google Scholar
  38. [38]
    X. Liu, K. L. Teo, and Y. Zhang, “Absolute stability of impulsive control systems with time delay,” Nonlinear Analysis: Theory, Methods & Applications vol. 62, no. 3, pp. 429–453, 2005.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Xiaodi Li
    • 1
    Email author
  • Wu-Hua Chen
    • 2
  • Wei Xing Zheng
    • 3
  • Qing-guo Wang
    • 4
  1. 1.School of Mathematics and StatisticsShandong Normal UniversityJi’nanP. R. China
  2. 2.College of Mathematics and Information ScienceGuangxi UniversityNanning, GuangxiP. R. China
  3. 3.School of Computing, Engineering and MathematicsWestern Sydney UniversitySydneyAustralia
  4. 4.Institute for Intelligent SystemsUniversity of JohannesburgJohannesburgSouth Africa

Personalised recommendations