Neural Computing and Applications

, Volume 31, Issue 11, pp 6933–6943 | Cite as

Positive solutions and exponential stability of positive equilibrium of inertial neural networks with multiple time-varying delays

  • Le Van HienEmail author
  • Le Dao Hai-An
Original Article


This paper is concerned with positive solutions and global exponential stability of positive equilibrium of inertial neural networks with multiple time-varying delays. By utilizing the comparison principle via differential inequalities, we first explore conditions on damping coefficients and self-excitation coefficients to ensure that, with nonnegative connection weights and inputs, all state trajectories of the system initiating in an admissible set of initial conditions are always nonnegative. Then, based on the method of using homeomorphisms, we derive conditions in terms of linear programming problems via M-matrices for the existence, uniqueness, and global exponential stability of a positive equilibrium of the system. Two examples with numerical simulations are given to illustrate the effectiveness of the obtained results.


Inertial neural networks Positive equilibrium Exponential stability Time-varying delay 


Compliance with ethical standards

Conflicts of interest

The authors declare that no potential conflict of interest to be reported to this work.


  1. 1.
    Soulié FF, Gallinari P (1998) Industrial applications of neural networks. World Scientific Publishing, SingaporeCrossRefGoogle Scholar
  2. 2.
    Venketesh P, Venkatesan R (2009) A survey on applications of neural networks and evolutionary techniques in web caching. IETE Tech Rev 26:171–180CrossRefGoogle Scholar
  3. 3.
    Mrugalski M, Luzar M, Pazera M, Witczak M, Aubrun C (2016) Neural network-based robust actuator fault diagnosis for a non-linear multi-tank system. ISA Trans 61:318–328CrossRefGoogle Scholar
  4. 4.
    Witczak P, Patan K, Witczak M, Mrugalski M (2017) A neural network approach to simultaneous state and actuator fault estimation under unknown input decoupling. Neurocomputing 250:65–75CrossRefGoogle Scholar
  5. 5.
    Kiakojoori S, Khorasani K (2016) Dynamic neural networks for gas turbine engine degradation prediction, health monitoring and prognosis. Neural Comput Appl 27:2157–2192CrossRefGoogle Scholar
  6. 6.
    Gong M, Zhao J, Liu J, Miao Q, Jiao J (2016) Change detection in synthesis aperture radar images based on deep neural networks. IEEE Trans Neural Netw Learn Syst 27:125–138MathSciNetCrossRefGoogle Scholar
  7. 7.
    Baldi P, Atiya AF (1995) How delays affect neural dynamics and learning. IEEE Trans Neural Netw 5:612–621CrossRefGoogle Scholar
  8. 8.
    Lu H (2012) Chaotic attractors in delayed neural networks. Phys Lett A 298:109–116CrossRefGoogle Scholar
  9. 9.
    Zhang H, Wang Z, Liu D (2014) A comprehensive review of stability analysis of continuous-time recurrent neural networks. IEEE Trans Neural Netw Learn Syst 25:1229–1262CrossRefGoogle Scholar
  10. 10.
    Liu B (2015) Pseudo almost periodic solutions for CNNs with continuously distributed leakage delays. Neural Process Lett 42:233–256CrossRefGoogle Scholar
  11. 11.
    Arik S (2016) Dynamical analysis of uncertain neural networks with multiple time delays. Int J Syst Sci 47:730–739MathSciNetCrossRefGoogle Scholar
  12. 12.
    Li L, Yang YQ, Lin G (2016) The stabilization of BAM neural networks with time-varying delays in the leakage terms via sampled-data control. Neural Comput Appl 27:447–457CrossRefGoogle Scholar
  13. 13.
    Liu B (2017) Global exponential convergence of non-autonomous SICNNs with multi-proportional delays. Neural Comput Appl 28:1927–1931CrossRefGoogle Scholar
  14. 14.
    Manivannan R, Samidurai R, Sriraman R (2017) An improved delay-partitioning approach to stability criteria for generalized neural networks with interval time-varying delays. Neural Comput Appl 28:3353–3369CrossRefGoogle Scholar
  15. 15.
    Hai-An LD, Hien LV, Loan TT (2017) Exponential stability of non-autonomous neural networks with heterogeneous time-varying delays and destabilizing impulses. Vietnam J Math 45:425–440MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lee TH, Trinh MH, Park JH (2017) Stability analysis of neural networks with time-varying delay by constructing novel Lyapunov functionals. IEEE Trans Neural Netw Learning Syst. CrossRefGoogle Scholar
  17. 17.
    Wheeler DW, Schieve WC (1997) Stability and chaos in an inertial two neuron system. Phys D Nonlin Phenom 105:267–284CrossRefGoogle Scholar
  18. 18.
    Koch C (1984) Cable theory in neurons with active linearized membrane. Biol Cybern 50:15–33CrossRefGoogle Scholar
  19. 19.
    Babcock KL, Westervelt RM (1986) Stability and dynamics of simple electronic neural networks with added inertia. Phys D Nonlin Phenom 23:464–469CrossRefGoogle Scholar
  20. 20.
    Tu Z, Cao J, Hayat T (2016) Matrix measure based dissipativity analysis for inertial delayed uncertain neural networks. Neural Netw 75:47–55CrossRefGoogle Scholar
  21. 21.
    Wan P, Jian J (2017) Global convergence analysis of impulsive inertial neural networks with time-varying delays. Neurocomputing 245:68–76CrossRefGoogle Scholar
  22. 22.
    Tu Z, Cao J, Alsaedi A, Alsaadi F (2017) Global dissipativity of memristor-based neutral type inertial neural networks. Neural Netw 88:125–133CrossRefGoogle Scholar
  23. 23.
    Zhang G, Zeng Z, Hu J (2018) New results on global exponential dissipativity analysis of memristive inertial neural networks with distributed time-varying delays. Neural Netw 97:183–191CrossRefGoogle Scholar
  24. 24.
    Ke Y, Miao C (2013) Stability analysis of inertial Cohen–Grossberg-type neural networks with time delays. Neurocomputing 117:196–205CrossRefGoogle Scholar
  25. 25.
    Zhang Z, Quan Z (2015) Global exponential stability via inequality technique for inertial BAM neural networks with time delays. Neurocomputing 151:1316–1326CrossRefGoogle Scholar
  26. 26.
    Cui N, Jiang H, Hu C, Abdurahman A (2018) Global asymptotic and robust stability of inertial neural networks with proportional delays. Neurocomputing 272:326–333CrossRefGoogle Scholar
  27. 27.
    Tu Z, Cao J, Hayat T (2016) Global exponential stability in Lagrange sense for inertial neural networks with time-varying delays. Neurocomputing 171:524–531CrossRefGoogle Scholar
  28. 28.
    Wang J, Tian L (2017) Global Lagrange stability for inertial neural networks with mixed time-varying delays. Neurocomputing 235:140–146CrossRefGoogle Scholar
  29. 29.
    He X, Huang TW, Yu JZ, Li CD, Li CJ (2017) An inertial projection neural network for solving variational inequalities. IEEE Trans Cybern 47:809–814CrossRefGoogle Scholar
  30. 30.
    Smith H (2008) Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. American Mathematical Society, ProvidenceCrossRefGoogle Scholar
  31. 31.
    Mózaryn J, Kurek JE (2010) Design of a neural network for an identification of a robot model with a positive definite inertia matrix. In: Artificial Intelligence and Soft Computing. Springer, BerlinGoogle Scholar
  32. 32.
    Ma GJ, Wu S, Cai GQ (2013) Neural networks control of the Ni-MH power battery positive mill thickness. Appl Mech Mater 411–414:1855–1858CrossRefGoogle Scholar
  33. 33.
    Lu W, Chen T (2007) \(R^n_+\)-global stability of a Cohen–Grossberg neural network system with nonnegative equilibria. Neural Netw 20:714–722Google Scholar
  34. 34.
    Liu B, Huang L (2008) Positive almost periodic solutions for recurrent neural networks. Nonlinear Anal Real World Appl 9:830–841MathSciNetCrossRefGoogle Scholar
  35. 35.
    Hien LV (2017) On global exponential stability of positive neural networks with time-varying delay. Neural Netw 87:22–2617CrossRefGoogle Scholar
  36. 36.
    He Y, Ji MD, Zhang CK, Wu M (2016) Global exponential stability of neural networks with time-varying delay based on free-matrix-based integral inequality. Neural Netw 77:80–86CrossRefGoogle Scholar
  37. 37.
    Arino O, Hbid ML, Ait Dads E (2002) Delay differential equations and applications. Springer, DordrechtGoogle Scholar
  38. 38.
    Forti M, Tesi A (1995) New conditions for global stability of neural networks with application to linear and quadratic programming problems. IEEE Trans Circuits Syst-I: Fund 42:354–366MathSciNetCrossRefGoogle Scholar
  39. 39.
    Hien LV, Son DT (2015) Finite-time stability of a class of non-autonomous neural networks with heterogeneous proportional delays. Appl Math Comput 251:14–23MathSciNetzbMATHGoogle Scholar
  40. 40.
    Haykin S (1999) Neural networks: a comprehensive foundation. Prentice Hall, Upper Saddle RiverzbMATHGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2018

Authors and Affiliations

  1. 1.Department of MathematicsHanoi National University of EducationHanoiVietnam
  2. 2.Department of MathematicsVietnam Maritime UniversityHai PhongVietnam

Personalised recommendations