DYNAMICS INVESTIGATION OF ONE WEAKLY NONLINEAR SYSTEM WITH DELAY ARGUMENT

Khusainov Denis Ya.Doctor in physics and mathematics, professor of Kiev National Taras Shevchenko University, Kiev and University of Technology, CEITEC, Brno (Czech Republic)

Diblik Jozef Doctor of sciences, docent of University of Technology, CEITEC, Brno (Czech Republic)

Bashtinec JaromirCandidate of sciences, docent of University of Technology, CEITEC, Brno (Czech Republic)

Shatyrko Andrey V.,Candidate in physics and mathematics, docent of Kiev National Taras Shevchenko University, Kiev

pages 22–37

DOI: 10.1615/JAutomatInfScien.v50.i1.20

A mathematical model of neural network dynamics represented by a system of differential equations with time-delay argument and an asymptotically stable linear part is considered. With using the direct Lyapunov method, sufficient conditions for asymptotic stability are obtained and exponential estimates of the decay of solutions are constructed. The results are formulated in the form of matrix algebraic inequalities (using LMI).

  1. McCulloch W.S., Pitts W., А logical calculus of the ideas immanent in nervous activity, Bulletin of Mathematica1 Biophysics, 1943, No. 5, 115–133.
  2. Haykin S., Neural networks: a comprehensive foundation, 2nd edition, Prentice Hall, New Jersey, 1998.
  3. Hopfield J.J., Neurons with graded response have collective computational properties like those of two state neurons, Proceedings of the National Academy of Sciences, 1984, No. 81, 3088–3092.
  4. Rall W., Cable theory for dendritic neurons. In Methods in Neuronal Modeling, MIТ Press, Cambridge, 1989, 9–62.
  5. Pineda F.J., Generalization of back propagation to rеccurent neural1 networks, Physical Review Letters, 1987, No. 59, 2229–2232.
  6. Scott А.С., Neurophysics, Wiley, New York, 1977.
  7. Gopalsamy K., Leakage Delays in BAM, Journal of Mathematical Analysis and Applications, 2007, No. 325, 1117–1132.
  8. Cohen М.А., Grossberg S., Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEЕ Transactions on Systems, Маn and Cybernetics, 1983, SMC-13, 815–826.
  9. Wu A., Zeng Z., Algebraical criteria of stability for delayed memristive neural networks, J. Advances in Difference Equations, 2015, 111, 12, https://doi.org/10.1186/s13662-015-0449-z
  10. Wanga X., Shea K., Zhong S., Cheng J., On extended dissipativity analysis for neural networks with time-varying delay and general activation functions, Ibid., 2016, 79, 16, https://doi.org/10.1186/s13662-016-0769-7
  11. Liu J., Xu R., Passivity analysis and state estimation for a class of memristor-based neural networks with multiple proportional delays, Ibid., 2017, 34, 20, https://doi.org/10.1186/ s13662-016-1069-y
  12. Arkhangelskiy V.I., Bogaenko I.N., Grabovskiy G.G., Ryumshin N.A., Neural networks in automatization systems [in Russian], Tekhnika, Kiev, 1999.
  13. Berezansky L., Idels L., Troib L., Global dynamics of the class on nonlinear nonautonomous systems with time-varying delays, Nonlinear Analysis, 2011, 74, No. 18, 7499–7512.
  14. Brokan E., Sadyrbaev F., On a differential system arising in the network control theory, Ibid., 2016, 21, No. 5, 687–701, http://dx.doi.org/10.15388/NA.2016.5.8
  15. Liang J., Cao J., Ho D.W.C., Discrete-time bidirectional associative memory neural networks with variable delays, Physics Letters, 2005, A 335, 226–234.
  16. Atslega S., Finaskins D., Sadyrbaev F., On a planar system arising in the network control theory, Mathematical Modelling and Analysis, 2016, 21, No. 3, 385–398, http://dx.doi.org/10.3846/ 13926292.216.1172131
  17. Sirenko A.S., Shakotko T.I., Khusainov D.Ya., On one approach to studying stability of neural networks model with delay using Lyapunov second method, Visnyk Kuivskogo Natsionalnogo Universytetu imeni Tarasa Shevchenko, 2014, 232–237.
  18. Khusainov D.Ya., Diblik I., Bashtinets Ya., Sirenko A.S., Nonuniform in delay stability of one weakly nonlinear system with aftereffect, Trudy Instituta Prikladnoy Matematiki i Mekhaniki, 2015, No. 29, 129–146.
  19. Aizerman M.A, Gantmaher F.R., Absolute stability of regulator systems, Holden-Day, San Francisco, 1964.
  20. Lur’e A.I., Some problems in the theory of automatic control, H.M. Stationary Office, London, 1957.
  21. Khusainov D.Ya., Shatyrko A.V., Method of Lyapunov functions in studying stability of differential functional systems [in Russian], Izdatelstvo Kievskogo Universiteta, Kiev, 1997.
  22. Khusainov D.Ya., Shatyrko A.V., Stability of nonlinear control systems with aftereffect [in Ukrainian], DP “Informatsiine analitychne agenstvo”, Kyiv, 2012.
  23. El’sgol’ts L.E., Norkin S.B., Introduction to the theory of the differential equations with deviating argument, Academic Press, New York, 1973.