Explicit criteria for exponential stability of nonlinear singular equations with delays

Original Paper


By a novel approach, we give some explicit criteria for global exponential stability of singular nonlinear differential equations with delays. An application to electrical networks containing lossless transmission lines is presented.


Singular system Nonlinear differential equations with delays Exponential stability 

Mathematics Subject Classification




The author would like to thank the anonymous reviewers for carefully reading the manuscript and some constructive suggestions. This work is supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant 101.01-2016.09.


  1. 1.
    Dai, L.: Singular Control Systems, Lecture Notes: in Control and Information Sciences, vol. 118 (1989)Google Scholar
  2. 2.
    Brayton, R.: Small signal stability criterion for electrical networks containing lossless transmission lines. IBM J. Res. Dev. 12, 431–440 (1968)CrossRefMATHGoogle Scholar
  3. 3.
    Brayton, R.: Nonlinear oscillations in a distributed network. Q. Appl. Math. 24, 289–301 (1967)CrossRefMATHGoogle Scholar
  4. 4.
    Jiemei, Z., Zhonghui, H.: Stability and passivity analysis for singular systems with time-varying delay. In: Proceedings of the 34th Chinese Control Conference, Hangzhou, China 294-298 (2015)Google Scholar
  5. 5.
    Fridman, E.: Stability of linear descriptor systems with delay: a Lyapunov-based approach. J. Math. Anal. Appl. 273, 24–44 (2002)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Halanay, A., Rasvan, V.: Stability radii for some propagation models. IMA J. Math. Control Inform. 14, 95–107 (1997)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Haidar, A., Boukas, E.K.: Exponential stability of singular systems with multiple time-varying delays. Automatica 45, 539–545 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kolmanovskii, V.B., Myshkis, A.: Applied Theory of Functional Differential Equations. Kluwer, Dordrecht, the Netherlands (1996)MATHGoogle Scholar
  9. 9.
    Ngoc, P.H.A., Trinh, H.: Novel criteria for exponential stability of linear neutral time-varying differential systems. IEEE Trans. Autom. Control 61, 1590–1594 (2016)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ngoc, P.H.A., Trinh, H.: Stability analysis of nonlinear neutral functional differential equations. SIAM J. Control Optim. 55, 3947–3968 (2017)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Niamsup, P., Phat, V.N.: A new result on finite-time control of singular linear time-delay systems. Appl. Math. Lett. 60, 1–7 (2016)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Phat, V.N., Sau, N.H.: On exponential stability of linear singular positive delayed systems. Appl. Math. Lett. 38, 67–72 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Slemrod, M.: Nonexistence of oscillations in a nonlinear distributed network. J. Math. Anal. Appl. 36, 22–40 (1971)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Wu, J., Lu, G., Wo, S., Xiao, X.: Exponential stability and stabilization for nonlinear descriptor systems with discrete and distributed delays. Int. J. Robust Nonlinear Control 23, 1393–1404 (2013)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsVietnam National University-HCMC, International UniversityThu Duc District, SaigonVietnam

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