Journal of Systems Science and Complexity

, Volume 32, Issue 2, pp 479–495 | Cite as

Novel Criteria for Exponential Stability of Linear Non-Autonomous Functional Differential Equations

  • Ngoc Pham Huu AnhEmail author
  • Tran Thai Bao
  • Tinh Cao Thanh
  • Huy Nguyen Dinh


General linear non-autonomous functional differential equations are considered. Explicit criteria for exponential stability are given. Furthermore, the authors present an explicit robust stability bound for systems subject to time-varying perturbations. Two examples are given to illustrate the obtained results.


Exponential stability linear functional differential equations perturbation 


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  1. [1]
    Haddad W M, Chellaboina V, and Hui Q, Nonnegative and Compartmental Dynamical Systems, Princeton University Press, Princeton, New Jersey, 2010.CrossRefzbMATHGoogle Scholar
  2. [2]
    Kuang Y, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191, Academic Press, New York, 1993.zbMATHGoogle Scholar
  3. [3]
    Smith H, An Introduction to Delay Differential Equations with Sciences Applications to the Life, Texts in Applied Mathematics, 57, Springer, New York, Dordrecht, Heidelberg, London, 2011.CrossRefGoogle Scholar
  4. [4]
    Dashkovskiy S and Naujok L, Lyapunov-Razumikhin and Lyapunov-Krasovskii theorems for interconnected ISS time-delay systems, Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS) 5–9 July, Budapest, Hungary, 2010, 1180–1184.Google Scholar
  5. [5]
    Dieudonné J, Foundations of Modern Analysis, Academic Press, New York, 1988.zbMATHGoogle Scholar
  6. [6]
    Driver R D, Existence and stability of solutions of a delay differential system, Archive for Rational Mechanics and Analysis, 1962, 10: 401–426.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Fridman E, New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems, Systems & Control Letters, 2001, 43: 309–319.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Fridman E, Stability of systems with uncertain delays: A new complete Lyapunov-Krasovskii functional, IEEE Transactions on Automatic Control, 2006, 51: 885–890.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Faria T and Oliveira J, Local and global stability for Votka-Volterra systems with distributed delays and instantaneous negative feedbacks, Journal of Differential Equations, 2008, 244: 1049–1079.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Idels L and Kipnis M, Stability criteria for a nonlinear non-autonomous system with delays, Applied Mathematical Modelling, 2009, 33: 2293–2297.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Liu X, Yu W, and Wang L, Stability analysis for continuous-time positive systems with timevarying delays, IEEE Transactions on Automatic Control, 2010, 55: 1024–1028.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Ngoc P H A, Strong stability radii of positive linear time-delay systems, International Journal of Robust and Nonlinear Control, 2005, 15: 459–472.MathSciNetCrossRefGoogle Scholar
  13. [13]
    Ngoc P H A, On positivity and stability of linear Volterra systems with delay, SIAM Journal on Control and Optimization, 2009, 48: 1939–1960.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Ngoc P H A, On exponential stability of nonlinear differential systems with time-varying delay, Applied Mathematics Letters, 2012, 25: 1208–1213.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Ngoc P H A, Stability of positive differential systems with delay, IEEE Transactions on Automatic Control, 2013, 58: 203–209.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Xueli S and Jigen P, A novel approach to exponential stability of nonlinear systems with timevarying delays, Journal of Computational and Applied Mathematics, 2011, 235: 1700–1705.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Wang F, Exponential asymptotic stability for nonlinear neutral systems with multiple delays, Nonlinear Analysis: Real World Applications, 2007, 8: 312–322.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Zhang B, Lam J, Xu S, et al., Absolute exponential stability criteria for a class of nonlinear time-delay systems, Nonlinear Analysis: Real World Applications, 2010, 11: 1963–1976.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Hale J and Lunel S M V, Introduction to Functional Differential Equations, Springer-Verlag Berlin, Heidelberg, New york, 1993.CrossRefzbMATHGoogle Scholar
  20. [20]
    Ngoc P H A and Hieu L T, New criteria for exponential stability of nonlinear difference systems with time-varying delay, International Journal of Control, 2013, 86: 1646–1651.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Son N K and Hinrichsen D, Robust stability of positive continuous-time systems, Numer. Funct. Anal. Optim., 1996, 17: 649–659.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Hewitt E and Stromberg K R, Real and Abstract Analysis, Springer-Verlag, New York, 1965.CrossRefzbMATHGoogle Scholar
  23. [23]
    Kolmanovskii V B and Nosov V R, Stability of Functional Differential Equations, Academic Press, New York, 1986.Google Scholar
  24. [24]
    Jiang M, Shen Y, and Liao X, On the global exponential stability for functional differential equations, Communications in Nonlinear Science and Numerical Simulation, 2005, 10: 705–713.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Kato J, Remarks on linear functional differential equations, Funkcialaj Ekvacioj, 1969, 12: 89–98.MathSciNetzbMATHGoogle Scholar
  26. [26]
    Ngoc P H A, Novel criteria for exponential stability of functional differential equations, Proceedings of the American Mathematical Society, 2013, 141: 3083–3091.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Editorial Office of JSSC & Springer-Verlag GmbH Germany 2019

Authors and Affiliations

  • Ngoc Pham Huu Anh
    • 1
    Email author
  • Tran Thai Bao
    • 2
  • Tinh Cao Thanh
    • 3
  • Huy Nguyen Dinh
    • 4
  1. 1.Department of MathematicsVietnam National University-HCMC, International University, Thu Duc DistrictSaigonVietnam
  2. 2.Faculty of Information SystemsVietnam National University-HCMC, University of Information Technology, Thu Duc districtSaigonVietnam
  3. 3.Department of MathematicsVietnam National University-HCMC, University of Information Technology, Thu Duc districtSaigonVietnam
  4. 4.Division of Applied Mathematics, Faculty of Applied SciencesVietnam National University-HCMC, Ho Chi Minh city University of Technology, Thu Duc districtSaigonVietnam

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