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International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Angheluta-Bauer, Luiza / Chen, Xi / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi / Yang, Xu

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Volume 20, Issue 2


Representation of Solutions and Finite Time Stability for Delay Differential Systems with Impulsive Effects

Zhongli You
  • Department of Mathematics, School of Mathematics and Statistics, Guizhou University, Guiyang, Guizhou 550025, P.R. China
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/ JinRong Wang
  • Corresponding author
  • Department of Mathematics, School of Mathematics and Statistics, Guizhou University, Guiyang, Guizhou 550025, P.R. China
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/ Yong Zhou
  • Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, P.R. China; Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
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/ Michal Fečkan
  • Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48 Bratislava, Slovakia, and Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia
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Published Online: 2019-01-18 | DOI: https://doi.org/10.1515/ijnsns-2018-0137


In this paper, we study finite time stability for linear and nonlinear delay systems with linear impulsive conditions and linear parts defined by permutable matrices. We introduce a new concept of impulsive delayed matrix function and apply the variation of constants method to seek a representation of solution of linear impulsive delay systems, which can be well used to deal with finite time stability. We establish sufficient conditions for the finite time stability results by using the properties of impulsive delayed matrix exponential and Gronwall’s integral inequalities. Finally, we give numerical examples to demonstrate the validity of theoretical results and present some possible advantage by comparing the current work with the previous literature.

Keywords: delay differential systems; impulsive effects; delayed exponential matrix; representation of solutions; finite time stability

MSC 2010: 34A37; 34D20


  • [1]

    D. D. Bainov and P. S. Simeonov, Theory of impulsive differential equations, series on advances in mathematics for applied sciences, vol. 28, World Scientific, Singapore, 1995.Google Scholar

  • [2]

    A. M. Samoilenko, N. A. Perestyuk and Y. Chapovsky, Impulsive differential equations, World Scientific series on nonlinear science, Series A, Monographs and treatises, 14 (1995).Google Scholar

  • [3]

    M. Benchohra, J. Henderson and S. K. Ntouyas, Impulsive differential equations and inclusions, Contemporary Mathematics and Its Applications, Hindawi Publishing Corporation, 2 (2006).Google Scholar

  • [4]

    M. U. Akhmet, J. Alzabut and A. Zafer, Perron’s theorem for linear impulsive differential equations with distributed delay, J. Comput. Appl. Math. 193 (2006), 204–218.Google Scholar

  • [5]

    J. Wang, M. Fečkan and Y. Zhou, A survey on impulsive fractional differential equations, Fract. Calc. Appl. Anal. 19 (2016), 806–831.Google Scholar

  • [6]

    J. Wang, M. Fečkan and Y. Zhou, Fractional order differential switched systems with coupled nonlocal initial and impulsive conditions, Bullet. Sci. Math. 141 (2017), 727–746.

  • [7]

    X. Li and J. Wu, Stability of nonlinear differential systems with state-dependent delayed impulses, Automatica. 64 (2016), 63–69.Google Scholar

  • [8]

    G. Suia and X. Li, et al., Sufficient conditions for pulse phenomena of nonlinear systems with state-dependent impulses, J. Nonlinear Sci. Appl. 9 (2016), 2649–2657.Google Scholar

  • [9]

    X. Li, M. Bohner and C. Wang, Impulsive differential equations: periodic solutions and applications, Automatica. 52 (2015), 173–178.Google Scholar

  • [10]

    X. Zhang and X. Li, Input-to-state stability of non-linear systems with distributed-delayed impulses, IET Control Theory Appl. 11 (2016), 81–89.

  • [11]

    X. Li, J. Shen and R. Rakkiyappan, Persistent impulsive effects on stability of functional differential equations with finite or infinite delay, Appl. Math. Comput. 329 (2018), 14–22.

  • [12]

    X. Li, J. Shen and H. Akca, et al., Comparison principle for impulsive functional differential equations with infinite delays and applications, Commun. Nonlinear Sci. Numer. Simul. 57 (2018), 309–321.Google Scholar

  • [13]

    X. Li, S. Song and J. Wu, Impulsive control of unstable neural networks with unbounded time-varying delays, Sci. China Inform. Sci. 61 (2018), 012203.Google Scholar

  • [14]

    X. Li and J. Wu, Sufficient stability conditions of nonlinear differential systems under impulsive control with state-dependent delay, IEEE Trans. Automat. Control. 63 (2018), 306–311.Google Scholar

  • [15]

    D. Ya. Khusainov and G. V. Shuklin, Linear autonomous time-delay system with permutation matrices solving, Stud. Univ. Žilina, 17 (2003), 101–108.

  • [16]

    J. Dibl&’ık and D. Ya. Khusainov, Representation of solutions of discrete delayed system x(k+1)=Ax(k)+Bx(k-m)+f(k) with commutative matrices, J. Math. Anal. Appl. 318 (2006), 63–76.

  • [17]

    J. Dibl&’ık and D. Ya. Khusainov, Representation of solutions of linear discrete systems with constant coefficients and pure delay, Adv. Difference Equ. 2006 (2006), 080825.Google Scholar

  • [18]

    Z. You and J. Wang, Stability of impulsive delay differential equations, J. App. Math. Comput. 56 (2018), 253–268.

  • [19]

    D. Ya. Khusainov and G. V. Shuklin, Relative controllability in systems with pure delay, Int. J. Appl. Math. 2 (2005), 210–221.

  • [20]

    M. Medveď, M. Pospišil and L. Škripková, Stability and the nonexistence of blowing-up solutions of nonlinear delay systems with linear parts defined by permutable matrices, Nonlinear Anal. 74 (2011), 3903–3911.

  • [21]

    M. Medveď and M. Pospišil, Sufficient conditions for the asymptotic stability of nonlinear multidelay differential equations with linear parts defined by pairwise permutable matrices, Nonlinear Anal. 75 (2012), 3348–3363.

  • [22]

    J. Dibl&’ık, M. Fečkan and M. Pospišil, Representation of a solution of the Cauchy problem for an oscillating system with two delays and permutable matrices, Ukrainian Math. J. 65 (2013), 58–69.Google Scholar

  • [23]

    J. Dibl&’ık, D. Ya. Khusainov and M. Růžičková, Controllability of linear discrete systems with constant coefficients and pure delay, SIAM J. Control Optim. 47 (2008), 1140–1149.

  • [24]

    J. Dibl&’ık, M. Fečkan and M. Pospišil, On the new control functions for linear discrete delay systems, SIAM J. Control Optim. 52 (2014), 1745–1760.Google Scholar

  • [25]

    J. Dibl&’ık and B. Morávková, Discrete matrix delayed exponential for two delays and its property, Adv. Diff. Equ. 2013 (2013), 1–18.Google Scholar

  • [26]

    J. Dibl&’ık and B. Morávková, Representation of the solutions of linear discrete systems with constant coefficients and two delays, Abstr. Appl. Anal. 2014 (2014), 1–19.Web of ScienceGoogle Scholar

  • [27]

    J. Dibl&’ık, D. Ya. Khusainov, J. Baštinec and A. S. Sirenko, Exponential stability of linear discrete systems with constant coefficients and single delay, Appl. Math. Lett. 51 (2016), 68–73.Google Scholar

  • [28]

    A. Boichuk, J. Dibl&’ık, D. Khusainov and M. Růžičková, Fredholm’s boundary-value problems for differential systems with a single delay, Nonlinear Anal. 72 (2010), 2251–2258.

  • [29]

    M. Posp&’ıšil, Representation and stability of solutions of systems of functional differential equations with multiple delays, Electron. J. Qual. Theory Differ. Equ. 54 (2012), 1–30.Google Scholar

  • [30]

    M. Posp&’ıšil, Representation of solutions of delayed difference equations with linear parts given by pairwise permutable matrices via Z-transform, Appl. Math. Comput. 294 (2017), 180–194.

  • [31]

    Z. Luo, W. Wei and J. Wang, On the finite time stability of nonlinear delay differential equations, Nonlinear Dyn. 89 (2017), 713–722.

  • [32]

    Z. You, J. Wang and D. O’Regan, Exponential stability and relative controllability of nonsingular delay systems, Bull Braz. Math. Soc. (2018), .CrossrefGoogle Scholar

  • [33]

    P. Dorato, Short time stability in linear time-varying systems, in: Proc. IRE Int. Convention Record, Part 4, (1961), 83–87.Google Scholar

  • [34]

    M. P. Lazarević, D. Debeljković and Z. Nenadić, Finite-time stability of delayed systems, IMA J. Math. Contr. Infor. 17 (2000), 101–109.Google Scholar

  • [35]

    F. Amato, M. Ariola and C. Cosentino, Robust finite-time stabilisation of uncertain linear systems, Int. J. Control. 84 (2011), 2117–2127.CrossrefWeb of ScienceGoogle Scholar

  • [36]

    M. P. Lazarević and A. M. Spasić, Finite-time stability analysis of fractional order time-delay system: Gronwall’s approach, Math. Comput. Model. 49 (2009), 475–481.Google Scholar

  • [37]

    L. Li, F. Meng and P. Ju, Some new integral inequalities and their applications in studying the stability of nonlinear integro differential equations with time delay, J. Math. Anal. Appl. 377 (2011), 853–862.

  • [38]

    Q. Feng, F. Meng and B. Zheng, Gronwall-Bellman type nonlinear delay integral inequalities on times scales, J. Math. Anal. Appl. 382 (2011), 772–784.Google Scholar

  • [39]

    M. Li and J. Wang, Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations, Appl. Math. Comput. 324 (2018), 254–265.

  • [40]

    X. Cao and J. Wang, Finite-time stability of a class of oscillating systems with two delays, Math. Meth. Appl. Sci. 41 (2018), 4943–4954.Web of ScienceCrossrefGoogle Scholar

  • [41]

    V. N. Phat, N. H. Muoi and M. V. Bulatov, Robust finite-time stability of linear differential-algebraic delay equations, Linear Algebra Appl. 487 (2015), 146–157.Google Scholar

  • [42]

    Y. Guo, Mean square global asymptotic stability of stochastic recurrent neural networks with distributed delays, Appl. Math. Comput. 215 (2009), 791–795.

  • [43]

    Z. Yuan, X. Yuan, F. Meng and H. Zhang, Some new delay integral inequalities and their applications, Appl. Math. Comput. 208 (2009), 231–237.Google Scholar

  • [44]

    M. Li and J. Wang, Representation of solution of a Riemann-Liouville fractional differential equation with pure delay, Appl. Math. Lett. 85 (2018), 118–124.

  • [45]

    Y. Guo, Globally Robust stability analysis for stochastic Cohen-Grossberg neural networks with impulse control and time-varying delays, Ukrainian Math. J. 69 (2018), 1220–1233.Web of ScienceCrossrefGoogle Scholar

  • [46]

    Z. You, J. Wang, D. O’Regan and Y. Zhou, Relative controllability of delay differential systems with impulses and linear parts defined by permutable matrices, Math. Meth. Appl. Sci. 42 (2019), 954–968.CrossrefWeb of ScienceGoogle Scholar

  • [47]

    J. G. Dong, Stability analysis of switched systems with general nonlinear disturbances, Math. Comput. Model. 58 (2013), 1563–1567.

  • [48]

    J. Shao and F. W. Meng, Gronwall-Bellman type inequalities and their applications to fractional differential equations, Abstr. Appl. Anal. 2013 (2013), 1056–1083.Web of ScienceGoogle Scholar

  • [49]

    D. D. Bainov and S. G. Hristova, Impulsive integral inequalities with a deviation of the argument, Math. Nachr. 171 (1995), 19–27.Google Scholar

  • [50]

    I. A. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problem of differential equation, Acta Math. Acad. Sci. Hungar. 7 (1956), 81–94.

About the article

Received: 2018-05-22

Accepted: 2018-12-16

Published Online: 2019-01-18

Published in Print: 2019-04-26

Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 20, Issue 2, Pages 205–221, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2018-0137.

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