Skip to main content
SpringerLink
Log in

Positive Solutions and Exponential Stability of Nonlinear Time-Delay Systems in the Model of BAM-Cohen-Grossberg Neural Networks

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

In this paper, the problems of positivity and exponential stability a BAM-Cohen-Grossberg neural networks model with time-varying delays and nonlinear self-excitation rates are studied. By novel comparison techniques via differential-integral inequalities, the exponential convergence of state trajectories to a unique positive equilibrium is established by tractable linear programming conditions, which can be effectively solved by various convex optimization algorithms. Numerical simulations are given to illustrate the effectiveness of the obtained theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Farina, L., Rinaldi, S.: Positive Linear Systems: Theory and Applications. John Wiley & Sons, New York (2000)

    Book  Google Scholar 

  2. Jacquez, J.: Compartmental Analysis in Biology and Medicine. University of Michigan Press, Ann Arbor, MI (1985)

    MATH  Google Scholar 

  3. Smith, H.: Monotone Dynamical System: An Introduction to the Theory of Competitive and Cooperative Systems. Amer Math Soc, Providence, USA (2008)

    Book  Google Scholar 

  4. Altafini, C.: Consensus problems on networks with antagonistic interactions. IEEE Trans. Autom. Control 58, 935–946 (2013)

    Article  MathSciNet  Google Scholar 

  5. Valcher, M.E., Zorzan, I.: State-feedback stabilization of multi-input compartmental systems. Syst. Control Lett. 119, 81–91 (2018)

    Article  MathSciNet  Google Scholar 

  6. Briat, C.: Stability and performance analysis of linear positive systems with delays using input-output methods. Int. J. Control 91, 1669–1692 (2018)

    Article  MathSciNet  Google Scholar 

  7. Mrugalski, M., et al.: Neural network-based robust actuator fault diagnosis for a non-linear multi-tank system. ISA Trans. 61, 318–328 (2016)

    Article  Google Scholar 

  8. Kiakojoori, S., Khorasani, K.: Dynamic neural networks for gas turbine engine degradation prediction, health monitoring and prognosis. Neural Comput. Appl. 27, 2157–2192 (2016)

    Article  Google Scholar 

  9. Gong, M., et al.: Change detection in synthetic aperture radar images based on deep neural networks. IEEE Trans. Neural Netw. Learn Syst. 27, 125–138 (2016)

    Article  MathSciNet  Google Scholar 

  10. Witczak, P., et al.: A neural network approach to simultaneous state and actuator fault estimation under unknown input decoupling. Neurocomputing 250, 65–75 (2017)

    Article  Google Scholar 

  11. Hien, L.V., Son, D.T.: Finite-time stability of a class of non-autonomous neural networks with heterogeneous proportional delays. Appl. Math. Comput. 251, 14–23 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Arik, S.: Dynamical analysis of uncertain neural networks with multiple time delays. Int. J. Syst. Sci. 47, 730–739 (2016)

    Article  MathSciNet  Google Scholar 

  13. Hai-An, L.D., Hien, L.V., Loan, T.T.: Exponential stability of non-autonomous neural networks with heterogeneous time-varying delays and destabilizing impulses. Vietnam J. Math. 45, 425–440 (2017)

    Article  MathSciNet  Google Scholar 

  14. Lee, T.H., Trinh, H., Park, J.H.: Stability analysis of neural networks with time-varying delay by constructing novel Lyapunov functionals. IEEE Trans. Neural. Netw. Learn. Syst. 29, 4238–4247 (2018)

    Article  Google Scholar 

  15. Ge, C., et al.: Robust passivity analysis for uncertain neural networks with discrete and distributed time-varying delays. Neurocomputing 364, 330–337 (2019)

    Article  Google Scholar 

  16. He, J., et al.: New \({\cal{H}}_\infty\) state estimation criteria of delayed static neural networks via the Lyapunov-Krasovskii functional with negative definite terms. Neural Netw. 123, 236–247 (2020)

    Article  Google Scholar 

  17. Mózaryn, J., Kurek, J.E.: Design of a neural network for an identification of a robot model with a positive definite inertia matrix. In: Artifical Intelligence and Soft Computing. Springer-Verlag, Berlin (2010)

  18. Ma, G.J., Wu, S., Cai, G.Q.: Neural networks control of the Ni-MH power battery positive mill thickness. Appl. Mech. Mater. 411–414, 1855–1858 (2013)

    Article  Google Scholar 

  19. Ameloot, T.J., den Bussche, J.V.: Positive neural networks in discrete time implement monotone-regular behaviors. Neural Comput. 27, 2623–2660 (2015)

    Article  MathSciNet  Google Scholar 

  20. Hien, L.V.: Positivity and Stability of Nonlinear Time-Delay Systems in Neural Networks. In: Park, J. (ed.) Recent Advances in Control Problems of Dynamical Systems and Networks. Springer, Cham (2021)

    Google Scholar 

  21. Hien, L.V.: On global exponential stability of positive neural networks with time-varying delay. Neural Netw. 87, 22–26 (2017)

    Article  Google Scholar 

  22. Hien, L.V., Hai-An, L.D.: Positive solutions and exponential stability of positive equilibrium of inertial neural networks with multiple time-varying delays. Neural Comput. Appl. 31, 6933–6943 (2019)

    Article  Google Scholar 

  23. Kosko, B.: Bidirectional associative memory. IEEE Trans. Syst. Man Cybern. 18, 49–60 (1988)

    Article  MathSciNet  Google Scholar 

  24. Hien, L.V., et al.: Existence and global asymptotic stability of positive periodic solution of delayed Cohen-Grossberg neural networks. Appl. Math. Comput. 240, 200–212 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Li, L., Yang, Y.Q., Lin, G.: The stabilization of BAM neural networks with time-varying delays in the leakage terms via sampled-data control. Neural Comput. Appl. 27, 447–457 (2016)

    Article  Google Scholar 

  26. Cohen, M.A., Grossberg, S.: Absolute stability of global pattern formation and parallel memory storage by competitive neural networks. IEEE Trans. Syst. Man Cybern. SMC-13, 815–826 (1983)

    Article  MathSciNet  Google Scholar 

  27. Mathiyalagan, K., Park, J.H., Sakthivel, R.: Synchronization for delayed memristive BAM neural networks using impulsive control with random nonlinearities. Appl. Math. Comput. 259, 967–979 (2015)

    MathSciNet  MATH  Google Scholar 

  28. Thipcha, J., Niamsup, P.: New exponential passivity of BAM neural networks with time-varying delays. Neural Comput. Appl. 29, 1593–1600 (2018)

    Article  Google Scholar 

  29. Aouiti, C., Assali, E.A.: Stability analysis for a class of impulsive bidirectional associative memory (BAM) neural networks with distributed delays and leakage time-varying delays. Neural Process. Lett. 50, 851–885 (2019)

    Article  Google Scholar 

  30. Ali, M.S., et al.: Asymptotic stability of Cohen-Grossberg BAM neutral type neural networks with distributed time varying delays. Neural Process. Lett. 46, 991–1007 (2017)

    Article  Google Scholar 

  31. Li, H., et al.: Fixed-time stabilization of impulsive Cohen-Grossberg BAM neural networks. Neural Netw. 98, 203–211 (2018)

    Article  Google Scholar 

  32. Hien, L.V., Hai-An, L.D.: Exponential stability of positive neural networks in bidirectional associative memory model with delays. Math. Meth. Appl. Sci. 42, 6339–6357 (2019)

    Article  MathSciNet  Google Scholar 

  33. Arino, O., Hbid, M.L., Ais Dads, E.: Delay Differential Equations and Applications. Springer, Dordrecht (2002)

    Google Scholar 

  34. Zeidler, E.: Nonlinear Functional Analysis and its Applications-I: Fixed point Theorems. Springer-Verlag, New York (1986)

    Book  Google Scholar 

  35. Plemmmons, R.J.: M-matrix characterizations: I-nonsingular M-matrices. Linear Alg Appl 18, 175–188 (1977)

    Article  MathSciNet  Google Scholar 

  36. Ngoc, P.H.A.: Stability of positive differential systems with delay. IEEE Trans. Autom. Control 58, 203–209 (2013)

    Article  MathSciNet  Google Scholar 

  37. Ngoc, P.H.A., Trinh, H.: Stability analysis of nonlinear neutral functional differential equations. SIAM J. Control. Optim. 55, 3947–3968 (2017)

    Article  MathSciNet  Google Scholar 

  38. Yang, G.: Exponential stability of positive recurrent neural networks with multi-proportional delays. Neural Process. Lett. 49, 67–78 (2019)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Fundamental Sciences, Hanoi University of Industry, Hanoi, Vietnam

    Le Thi Hong Dzung

  2. Department of Mathematics, Hanoi National University of Education, Hanoi, Vietnam

    Le Van Hien

Authors
  1. Le Thi Hong Dzung
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Le Van Hien
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Le Van Hien.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dzung, L.T.H., Van Hien, L. Positive Solutions and Exponential Stability of Nonlinear Time-Delay Systems in the Model of BAM-Cohen-Grossberg Neural Networks. Differ Equ Dyn Syst (2022). https://doi.org/10.1007/s12591-022-00605-y

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12591-022-00605-y

Keywords

Mathematics Subject Classification

Search

Navigation