Advertisement

Introduction

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

Abstract

Impulsive systems (systems of impulsive differential equations) model real world processes that undergo abrupt changes (impulses) in the state at a sequence of discrete times. These abrupt changes in systems’ states inspire the impulsive control mechanism. The theory of impulsive differential equations and its applications to impulsive control problems has been an active research area since 1990s. On the other hand, time-delay systems have been intensively studied in the past decades, mainly due to the ubiquity of time delays in physical processes such as proliferation process for solid avascular tumour, scattering process, milling process, and temperature control. Stability is one of the fundamental issues in system design, analysis and control. Recently, impulsive control has been shown to be a powerful approach to stabilize time-delay systems, and various stability and stabilization results have been obtained for impulsive time-delay systems.

References

  1. 2.
    H. Akca, V. Covachev, Z. Covacheva, Discrete-time counterparts of impulsive Hop field neural networks with leakage delays, in Springer Proceedings in Mathematics and Statistics: Differential and Difference Equations with Applications, vol. 47, pp. 351–358 (2013)zbMATHGoogle Scholar
  2. 4.
    F.M. Atici, D.C. Biles, A. Lebedinsky, An application of time scales to economics. Math. Comput. Model. 43, 718–726 (2006)MathSciNetzbMATHGoogle Scholar
  3. 9.
    Z. Bartosiewicz, E. Pawluszewicz, Realizations of linear control systems on timescales. Control Cybern. 35(4), 769–786 (2006)zbMATHGoogle Scholar
  4. 10.
    Z. Bartosiewicz, E. Pawluszewicz, Realizations of nonlinear control systems on timescales. IEEE Trans. Autom. Control 53(2), 571–575 (2008)zbMATHGoogle Scholar
  5. 14.
    M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications (Birkhäuser, Boston, 2001)zbMATHGoogle Scholar
  6. 15.
    M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales (Birkhäuser, Boston, 2003)zbMATHGoogle Scholar
  7. 20.
    A. Chen, D. Du, Global exponential stability of delayed BAM networks on timescales. Neurocomputing 71, 3582–3588 (2008)Google Scholar
  8. 21.
    W.H. Chen, W.X. Zeng, Exponential stability of nonlinear time delay systems with delayed impulse effects. Automatica 47(5), 1075–1083 (2011)MathSciNetzbMATHGoogle Scholar
  9. 23.
    S.K. Choi, N.J. Koo, On the stability of linear dynamic systems on time scales. J. Differ. Equ. Appl. 15(2), 167–183 (2009)MathSciNetzbMATHGoogle Scholar
  10. 25.
    S. Dashkovskiya, M. Kosmykovb, A. Mironchenkob, L. Naujok, Stability of interconnected impulsive systems with and without time delays, using Lyapunov methods. Nonlinear Anal. Hybrid Syst. 6(3), 899–915 (2012)MathSciNetGoogle Scholar
  11. 29.
    M. di Bernardo, A. Salvi, S. Santini, Distributed consensus strategy for platooning of vehicles in the presence of time-varying heterogeneous communication delays. IEEE Trans. Intell. Transp. Syst. 16(1), 102–112 (2015)Google Scholar
  12. 30.
    T.S. Doana, A. Kalauch, S. Siegmunda, F.R. Wirthb, Stability radii for positive linear time-invariant systems on time scales. Syst. Control Lett. 59(3–4), 173–179 (2010)MathSciNetGoogle Scholar
  13. 32.
    T. Faira, M.C. Gadotti, J.J. Oliveira, Stability results for impulsive functional differential equations with infinite delay. Nonlinear Anal. Theory, Methods Appl. 75(18), 6570–6587 (2012)Google Scholar
  14. 34.
    U. Fory, M. Bodnar, Time delays in proliferation process for solid avascular tumor. Math. Comput. Model. 37(11), 1201–1209 (2003)zbMATHGoogle Scholar
  15. 40.
    G.S. Guiseinov, E. Ozyylmaz, Tangent lines of generalized regular curves parameterized by time scales. Turk. J. Math. 25, 553–562 (2001)Google Scholar
  16. 45.
    S. Hilger, Ein Maβkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. PhD. Thesis, Universität Würzburg (1988, in German)Google Scholar
  17. 46.
    S. Hong, Stability criteria for set dynamic equations on time scales. Comput. Math. Appl. 59(11), 3444–3457 (2010)MathSciNetzbMATHGoogle Scholar
  18. 50.
    H.S. Hurd, J.B. Kaneene, J.W. Lloyd, A stochastic distributed-delay model of disease processes in dynamic populations. Prev. Vet. Med. 16(1), 21–29 (1993)Google Scholar
  19. 54.
    A. Khadra, X. Liu, X. Shen, Analyzing the robustness of impulsive synchronization coupled by linear delayed impulses. IEEE Trans. Autom. Control 4(4), 923–928 (2009)MathSciNetzbMATHGoogle Scholar
  20. 55.
    V. Kolmanovskii, A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations (Kluwer Academic Publishers, Dordrecht, 1999)zbMATHGoogle Scholar
  21. 58.
    V. Lakshmikantham, X. Liu, Stability Analysis in Terms of Two Measures (World Scientific, River Edge, 1993)zbMATHGoogle Scholar
  22. 59.
    V. Lakshmikantham, S. Sivasundaram, B. Kaymakcalan, Dynamic Systems on Measure Chains (Kluwer Academic Publishers, Boston, 1996)zbMATHGoogle Scholar
  23. 60.
    V. Lakshmikantham, A.S. Vatsala, Hybrid systems on time scales. J. Comput. Appl. Math. 141(1–2), 227–235 (2002)MathSciNetzbMATHGoogle Scholar
  24. 62.
    J. Li, J. Shen, New comparison results for impulsive functional differential equations. Appl. Math. Lett. 23(4), 487–493 (2010)MathSciNetzbMATHGoogle Scholar
  25. 63.
    X. Li, S. Song, Stabilization of delay systems: delay-dependent impulsive control. IEEE Trans. Autom. Control 62(1), 406–411 (2017)MathSciNetzbMATHGoogle Scholar
  26. 64.
    X. Li, J. Wu, Stability of nonlinear differential systems with state-dependent delayed impulses. Automatica 64, 63–69 (2016)MathSciNetzbMATHGoogle Scholar
  27. 66.
    X. Li, X. Zhang, S. Song, Effect of delayed impulses on input-to-state stability of nonlinear systems. Automatica 76, 378–382 (2017)MathSciNetzbMATHGoogle Scholar
  28. 67.
    Y. Li, X. Chen, L. Zhao, Stability and existence of periodic solutions to delayed Chhen-Grossberg BAM neural networks with impulses on time scales. Neurocomputing 72, 1621–1630 (2008)Google Scholar
  29. 68.
    B. Liu, D.J. Hill, Comparison principle and stability of discrete-time impulsive hybrid systems. IEEE Trans. Circuits Syst. I: Regul. Pap. 56(1), 233–245 (2009)MathSciNetGoogle Scholar
  30. 69.
    B. Liu, D.J. Hill, Uniform stability of large-scale delay discrete impulsive systems. Int. J. Control 82(2), 228–240 (2009)MathSciNetzbMATHGoogle Scholar
  31. 70.
    B. Liu, D.J. Hill, Uniform stability and ISS of discrete-time impulsive hybrid systems. Nonlinear Anal. Hybrid Syst. 4(2), 319–333 (2010)MathSciNetzbMATHGoogle Scholar
  32. 75.
    J. Liu, X. Liu, W.C. Xie, Input-to-state stability of impulsive and switching hybrid systems with time-delay. Automatica 47(5), 899–908 (2011)MathSciNetzbMATHGoogle Scholar
  33. 77.
    X. Liu, S. Shen, Y. Zhang, Q. Wang, Stability criteria for impulsive systems with time delay and unstable system matrices. IEEE Trans. Circuits Syst. I Regul. Pap. 54(10), 2288–2298 (2007)MathSciNetzbMATHGoogle Scholar
  34. 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)MathSciNetGoogle Scholar
  35. 80.
    X. Liu, K. Zhang, Existence, uniqueness and stability results for functional differential equations on time scales. Dyn. Syst. Appl. 25(4), 501–530 (2016)MathSciNetzbMATHGoogle Scholar
  36. 81.
    X. Liu, K. Zhang, Stabilization of nonlinear time-delay systems: distributed-delay dependent impulsive control. Syst. Control Lett. 120, 17–22 (2018)MathSciNetGoogle Scholar
  37. 82.
    X. Liu, K. Zhang, W.C. Xie, Synchronization of linear dynamical networks on time scales: pinning control via delayed impulses. Automatica 72, 147–152 (2016)MathSciNetzbMATHGoogle Scholar
  38. 83.
    X. Liu, K. Zhang, W.C. Xie, Stabilization of time-delay neural networks via delayed pinning impulses. Chaos Solitions Fractals 93, 223–234 (2016)MathSciNetzbMATHGoogle Scholar
  39. 84.
    X. Liu, K. Zhang, W.C. Xie, Consensus seeking in multi-agent systems via hybrid protocols with impulse delays. Nonlinear Anal. Hybrid Syst. 25, 90–98 (2017)MathSciNetzbMATHGoogle Scholar
  40. 85.
    X. Liu, K. Zhang, W.C. Xie, Pinning impulsive synchronization of reaction-diffusion neural networks with time-varying delays. IEEE Trans. Neural Netw. Learn. Syst. 28(5), 1055–1067 (2017)Google Scholar
  41. 86.
    X. Liu, K. Zhang, W.C. Xie, Consensus of multi-agent systems via hybrid impulsive protocols with time-delay. Nonlinear Anal. Theory Methods Appl. 30, 134–146 (2018)MathSciNetGoogle Scholar
  42. 87.
    X. Liu, K. Zhang, W.C. Xie, Impulsive consensus of networked multi-agent systems with distributed delays in agent dynamics and impulsive protocols. J. Dyn. Syst. Meas. Control (2018). https://doi.org/10.1115/1.4041202 Google Scholar
  43. 88.
    X. Liu, Z. Zhang, Uniform asymptotic stability of impulsive discrete systems with time delay. Nonlinear Anal. Theory Methods Appl. 74(15), 4941–4950 (2011)MathSciNetzbMATHGoogle Scholar
  44. 90.
    X.H. Long, B. Balachandran, B.P. Mann, Dynamics of milling processes with variable time delays. Nonlinear Dyn. 47(1–3), 49–63 (2007)zbMATHGoogle Scholar
  45. 101.
    Y. Ma, J. Sun, Stability criteria of delayed impulsive systems on time scales. Nonlinear Anal. Theory Methods Appl. 67(4), 1181–1189 (2007)zbMATHGoogle Scholar
  46. 102.
    Y. Ma, J. Sun, Stability criteria of impulsive systems on time scales. J. Comput. Appl. Math. 213(2), 400–407 (2008)MathSciNetzbMATHGoogle Scholar
  47. 107.
    S. Mohamad, K. Gopalsamy, Exponential stability of continuous-time and discrete-time cellular neural networks with delays. Appl. Math. Comput. 135(1), 17–38 (2003)MathSciNetzbMATHGoogle Scholar
  48. 109.
    A.A. Movchan, Stability of processes with respect to two matrices. Prikladnaya Matematika i Mekhanika 24, 988–1001 (1960)Google Scholar
  49. 113.
    S. Pan, J. Sun, S. Zhao, Robust filtering for discrete time piecewise impulsive systems. Signal Process. 90(1), 324–330 (2010)zbMATHGoogle Scholar
  50. 114.
    S. Peng, Y. Zhang, Razumikhin-type theorems on pth moment exponential stability of impulsive stochastic delay differential equations. IEEE Trans. Autom. Control 55(8), 1917–1922 (2010)zbMATHGoogle Scholar
  51. 115.
    C. Pozsche, S. Siegmund, F. Wirth, A spectral characterization of exponential stability for linear time-invariant systems on time scales. Discrete Contin. Dynam. Syst. A 9(5), 1223–1241 (2003)MathSciNetzbMATHGoogle Scholar
  52. 119.
    L. Salvadori, Some contributions to asymptotic stability theory. Annales de la Societé Scientifique de Bruxelles 88, 183–194 (1974)MathSciNetzbMATHGoogle Scholar
  53. 120.
    G.P. Samanta, Permanence and extinction of a nonautonomous HIV/AIDS epidemic model with distributed time delay. Nonlinear Anal. Real World Appl. 12(2), 1163–1177 (2011)MathSciNetzbMATHGoogle Scholar
  54. 121.
    M. Sassoli de Bianchi, Time-delay of classical and quantum scattering processes: a conceptual overview and a general definition. Centr. Eur. J. Phys. 10(2), 282–319 (2012)Google Scholar
  55. 123.
    I.M. Stamova, Impulsive control for stability of n-species Lotka-Volterra cooperation models with finite delays. Appl. Math. Lett. 23(9), 1003–1007 (2010)MathSciNetzbMATHGoogle Scholar
  56. 132.
    P. Wang, M. Wu, On the ϕ 0-stability of impulsive dynamic system on time scales. Electron. J. Differ. Equ. 128, 1–7 (2005)MathSciNetGoogle Scholar
  57. 133.
    P. Wang, M. Wu, Practical ϕ 0-stability of impulsive dynamic systems on time scales. Appl. Math. Lett. 20(6), 651–658 (2007)MathSciNetzbMATHGoogle Scholar
  58. 136.
    Q. Wang, X. Liu, Exponential stability of impulsive cellular neural networks with time delay via Lyapunov functionals. Appl. Math. Comput. 194(1), 186–198 (2007)MathSciNetzbMATHGoogle Scholar
  59. 137.
    Q. Wang, Q. Zhu, Razumikhin-type stability criteria for differential equations with delayed impulses. Electron. J. Qual. Theory Differ. Equ. 14, 1–18 (2013)MathSciNetzbMATHGoogle Scholar
  60. 140.
    K. Wu, X. Ding, Impulsive stabilization of delay difference equations and its application in Nicholson’s blowflies model. Adv. Differ. Equ. 2012(88) (2012). https://doi.org/10.1186/1687-1847-2012-88
  61. 142.
    S. Wu, C. Li, X. Liao, S. Duan, Exponential stability of impulsive discrete systems with time delay and applications in stochastic neural networks: a Razumikhin approach. Neurocomputing 82(1), 29–36 (2012)Google Scholar
  62. 148.
    Z. Yang, D. Xu, Stability analysis and design of impulsive control systems with time delay. IEEE Trans. Autom. Control 52(8), 1448–1454 (2007)MathSciNetzbMATHGoogle Scholar
  63. 150.
    D. Zennaro, A. Ahmad, L. Vangelista, E. Serpedin, H. Nounou, M. Nounou, Network-wide clock synchronization via message passing with exponentially distributed link delays. IEEE Trans. Commun. 61(5), 2012–2024 (2013)Google Scholar
  64. 151.
    Z. Zhan, W. Wei, On existence of optimal control governed by a class of the first-order linear dynamic systems on time scales. Appl. Math. Comput. 215, 2070–2081 (2009)MathSciNetzbMATHGoogle Scholar
  65. 153.
    K. Zhang, Stability and control of impulsive systems on time scales. PhD Thesis, Shandong University (2013)Google Scholar
  66. 154.
    K. Zhang, Impulsive control of dynamical networks. PhD Thesis, University of Waterloo (2017)Google Scholar
  67. 155.
    K. Zhang, X. Liu, Impulsive control of a class of discrete chaotic systems with parameter uncertainties, in The Proceedings of the 8th World Congress on Intelligent Control and Automation, 7–9 July 2010, Jinan, pp. 3691–3695 (2010)Google Scholar
  68. 156.
    K. Zhang, X. Liu, Controllability and observability of linear time-varying impulsive systems on time scales. AIP Conf. Proc. 1368(1), 25–28 (2011)Google Scholar
  69. 157.
    K. Zhang, X. Liu, Stability in terms of two measures for nonlinear impulsive systems on time scales by comparison method. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 19(2), 145–176 (2012)MathSciNetzbMATHGoogle Scholar
  70. 158.
    K. Zhang, X. Liu, Stability in terms of two measures for nonlinear impulsive systems on time scales. J. Appl. Math. Article ID 313029, 12 pp. (2013)Google Scholar
  71. 159.
    K. Zhang, X. Liu, Global exponential stability of nonlinear impulsive discrete systems with time delay, in The Proceedings of the 25th Chinese Control and Decision Conference, 25–27 May 2013, Guiyang, pp. 148–153 (2013)Google Scholar
  72. 160.
    K. Zhang, X. Liu, X.C. Xie, Global exponential stability of discrete-time delay systems subject to impulsive perturbations, in The Proceedings of the 4th International Conference on Complex Systems and Applications, 23–26 June 2014, Le Havre, pp. 239–244 (2014)Google Scholar
  73. 161.
    K. Zhang, X. Liu, W.C. Xie, Impulsive control and synchronization of spatiotemporal chaos in the Gray-Scott Model, in Interdisciplinary Topics in Applied Mathematics, Modeling and Computational Science, ed. by M. Cojocaru, I. Kotsireas, R. Makarov, R. Melnik, H. Shodiev. Springer Proceedings in Mathematics and Statistics, vol. 117 (Springer, Cham, 2015), pp. 549–555Google Scholar
  74. 162.
    K. Zhang, X. Liu, W.C. Xie, Pinning stabilization of cellular neural networks with time-delay via delayed impulses, in Mathematical and Computational Approaches in Advancing Modern Science and Engineering, ed. by J. Bélair, I. Frigaard, H. Kunze, R. Makarov, R. Melnik, R. Spiteri (Springer, Cham, 2016), pp. 763–773Google Scholar
  75. 164.
    Y. Zhang, Exponential stability of impulsive discrete systems with time delays. Appl. Math. Lett. 26(12), 2290–2297 (2012)MathSciNetzbMATHGoogle Scholar
  76. 166.
    Y. Zhang, J. Sun, G. Feng, Impulsive control of discrete systems with time delay. IEEE Trans. Autom. Control 54(4), 830–834 (2009)MathSciNetGoogle Scholar
  77. 167.
    Z. Zhang, Rubost H control of a class of discrete impulsive switched systems. Nonlinear Anal. Theory Methods Appl. 71(12), e2790–e2796 (2009)zbMATHGoogle Scholar
  78. 168.
    Z. Zhang, X. Liu, Robust stability of uncertain discrete impulsive switching systems. Comput. Math. Appl. 58(2), 380–389 (2009)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

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

Personalised recommendations