The Journal of Analysis

, Volume 27, Issue 1, pp 179–196 | Cite as

Stochastic stability of mode-dependent Markovian jump inertial neural networks

  • R. KrishnasamyEmail author
  • Raju K. George
Original Research Paper


The problem of stochastic stability analysis for Markovian jump inertial neural networks with mode-dependent time-varying delay is considered in this paper. The time-delay is assumed to be mode-dependent and time-varying. Using the suitable transformation technique, the second-order inertial neural network model is transformed into a system of first-order differential equations model. Delay-dependent stochastic stability condition is formulated in terms of linear matrix inequalities through the construction of Lyapunov–Krasovskii functional candidate involving mode-dependent time-varying delay. An integral inequality technique is used in the process to find the bounds of some integral terms. Finally, a numerical example is illustrated to show the effectiveness of the derived theoretical results.


Inertial neural networks Stochastic stability Linear matrix inequality Lyapunov–Krasovskii functional Markovian jump Mode-dependent delay 

Mathematics Subject Classification

34K20 93E15 70J25 



The work of the first author (R. Krishnasamy) is supported by the Science & Engineering Research Board (SERB), Department of Science & Technology, Government of India, New Delhi for the financial assistance through National Post-Doctoral Fellowship scheme (file No. PDF/2016/002992 dated 01/04/2017). Authors are very much thankful to the Editor-in-chief, Associate editor and anonymous reviewers for their careful reading, constructive comments and fruitful suggestions to improve the quality of this manuscript.

Compliance with ethical standards

Conflict of Interest

The authors declare that they have no conflict of interest.


  1. 1.
    Ashmore, J.F., and D. Attwell. 1985. Models for electrical tuning in hair cells. Proceedings of the Royal Society of London Series B Biological Sciences 226: 325–344.CrossRefGoogle Scholar
  2. 2.
    Babcock, K.L., and R.M. Westervelt. 1986. Stability and dynamics of simple electronic neural networks with added inertia. Physica D 23: 464–469.CrossRefGoogle Scholar
  3. 3.
    Badcock, K.L., and R.M. Westervelt. 1987. Dynamics of simple electronic neural networks. Physica D 28: 305–316.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Balasubramaniam, P., R. Krishnasamy, and R. Rakkiyappan. 2012. Delay-dependent stability criterion for a class of non-linear singular Markovian jump systems with mode-dependent interval time-varying delays. Communications in Nonlinear Science and Numerical Simulation 17: 3612–3627.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cui, N., H. Jiang, C. Hu, and A. Abdurahman. 2018. Global asymptotic and robust stability of inertial neural networks with proportional delays. Neurocomputing 272: 326–333.CrossRefGoogle Scholar
  6. 6.
    Gu, K., L. Kharitonov, and J. Chen. 2003. Stability of time delay systems. Boston: Birkhäuser.CrossRefzbMATHGoogle Scholar
  7. 7.
    Hale, J.K., and S.M. Verduyn Lunel. 1993. Introduction to functional differential equations. New York: Springer.CrossRefzbMATHGoogle Scholar
  8. 8.
    Ji, H., H. Zhang, and T. Senping. 2017. Reachable set estimation for inertial Markov jump BAM neural network with partially unknown transition rates and bounded disturbances. Journal of the Franklin Institute 354: 7158–7182.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lakshmanan, M., and D.V. Senthilkumar. 2010. Dynamics of nonlinear time-delay systems. Berlin: Springer.zbMATHGoogle Scholar
  10. 10.
    Lakshmanan, S., C.P. Lim, M. Prakash, S. Nahavandi, and P. Balasubramaniam. 2017. Neutral-type of delayed inertial neural networks and their stability analysis using the LMI approach. Neurocomputing 230: 243–250.CrossRefGoogle Scholar
  11. 11.
    Lee, T.H., and J.H. Park. 2017. A novel Lyapunov functional for stability of time-varying delay systems via matrix-refined-function. Automatica 80: 239–242.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Liu, Q., X. Liao, Y. Liu, S. Zhou, and S. Guo. 2009. Dynamics of an inertial two-neuron system with time delay. Nonlinear Dynamics 58: 573–609.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Liu, Y., W. Liu, L.M.A. Obaid, and I.A. Abbas. 2016. Exponential stability of Markovian jumping Cohen-Grossberg neural networks with mixed mode-dependent time-delays. Neurocomputing 177: 409–415.CrossRefGoogle Scholar
  14. 14.
    Ma, Y., and Y. Zheng. 2015. Synchronization of continuous-time Markovian jumping singular complex networks with mixed mode-dependent time delays. Neurocomputing 156: 52–59.CrossRefGoogle Scholar
  15. 15.
    Mahmoud, M.S., and P. Shi. 2003. Methodologies for control of jump time-delay systems. Dordrecht: Kluwer Academic Publishers.zbMATHGoogle Scholar
  16. 16.
    Muthukumar, P., and K. Subramanian. 2016. Stability criteria for Markovian jump neural networks with mode-dependent additive time-varying delays via quadratic convex combination. Neurocomputing 205: 75–83.CrossRefGoogle Scholar
  17. 17.
    Prakash, M., P. Balasubramaniam, and S. Lakshmanan. 2016. Synchronization of Markovian jumping inertial neural networks and its applications in image encryption. Neural Networks 83: 86–93.CrossRefGoogle Scholar
  18. 18.
    Qi, W., J.H. Park, J. Cheng, Y. Kao, and X. Gao. 2017. Anti-windup design for stochastic Markovian switching systems with mode-dependent time-varying delays and saturation nonlinearity. Nonlinear Analysis: Hybrid Systems 26: 201–211.MathSciNetzbMATHGoogle Scholar
  19. 19.
    Rakkiyappan, R., S. Premalatha, A. Chandrasekar, and J. Cao. 2016. Stability and synchronization analysis of inertial memristive neural networks with time delays. Cognitive Neurodynamics 10: 437–451.CrossRefGoogle Scholar
  20. 20.
    Sun, Z., and S.S. Ge. 2011. Stability theory of switched dynamical systems. London: Springer.CrossRefzbMATHGoogle Scholar
  21. 21.
    Tu, Z., J. Cao, A. Alsaedi, and F. Alsaedi. 2017. Global dissipativity of memristor-based neutral type inertial neural networks. Neural Networks 88: 125–133.CrossRefGoogle Scholar
  22. 22.
    Wang, J., H. Zhang, Z. Wang, and Z. Liu. 2017. Sampled-data synchronization of Markovian coupled neural networks with mode delays based on mode-dependent LKF. IEEE Transactions on Neural Networks and Learning Systems 28: 2626–2637.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Zhang, W., T. Huang, C. Li, J. Yang. 2017. Robust stability of inertial BAM neural networks with time delays and uncertainties via impulsive effect. Neural Processing Letters 2017:1–12. Google Scholar

Copyright information

© Forum D'Analystes, Chennai 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Space Science and TechnologyThiruvananthapuramIndia

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