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Asymptotical Stability for a Class of Complex-Valued Projective Neural Network

  • Jin-dong Li
  • Nan-jing Huang
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Abstract

In this paper, a new class of complex-valued projective neural network is introduced and studied on a nonempty, closed, and convex subset of a finite-dimensional complex space. An existence and uniqueness result for the equilibrium point of complex-valued projective neural network is proved under some suitable conditions. Moreover, by utilizing the linear matrix inequality technique, some sufficient conditions are presented to ensure the asymptotical stability of the complex-valued projective neural network. Finally, two examples are given to illustrate the validity and feasibility of main results.

Keywords

Complex-valued projective neural network Equilibrium point Linear matrix inequality technique Homeomorphism method Asymptotical stability 

Mathematics Subject Classification

49J40 34K20 92B20 
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Notes

Acknowledgements

The authors are grateful to the editor and the referees for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (11471230, 11671282) and the Applied Basic Research Programs of Department of Science and Technology of Sichuan Province of China (No. 2016JY0249).

References

  1. 1.
    Cao, J.D., Wan, Y.: Matrix measure strategies for stability and synchronization of inertial BAM neural network with time delays. Neural Netw. 53, 165–172 (2014)CrossRefzbMATHGoogle Scholar
  2. 2.
    Chua, L.O., Yang, L.: Cellular neural networks: theory. IEEE Trans. Circ. Syst. 35, 1257–1272 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Li, S., Wu, X., Zhang, J., Yang, R.: Asymptotical stability of Riemann–Liouville fractional neutral systems. Appl. Math. Lett. 69, 168–173 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Li, X.D.: Global exponential stability for a class of neural networks. Appl. Math. Lett. 22, 1235–1239 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Song, Q.K.: Exponential stability of recurrent neural networks with both time-varying delays and general activation functions via LMI approach. Neurocomputing 71, 2823–2830 (2008)CrossRefGoogle Scholar
  6. 6.
    Schreier, P.J., Scharf, L.L.: Statistical Signal Processing of Complex-Valued Data: The Theory of Improper and Noncircular Signals. Cambridge University Press, London (2010)CrossRefGoogle Scholar
  7. 7.
    Hirose, A.: Complex-Valued Neural Networks. Springer, Heidelberg (2006)CrossRefzbMATHGoogle Scholar
  8. 8.
    Amin, M.F., Amin, M.I., Al-Nuaimi, A.Y.H., Murase,K.: Wirtinger calculus based gradient descent and Levenberg–Marquardt learning algorithms in complex-valued neural networks. In: Lu, B.L., Zhang, L., Kwok, J.(eds.) ICONIP 2011, Part I, LNCS 7062, pp. 550–559 (2011)Google Scholar
  9. 9.
    Chen, X.F., Song, Q.K., Liu, Y.R., Zhao, Z.J.: Global \(\mu \)-stability of impulsive complex-valued neural networks with leakage delay and mixed delay delays. Abstr. Appl. Anal. Article ID 397532 (2014)Google Scholar
  10. 10.
    Chen, X.F., Zhao, Z.J., Song, Q.K., Hu, J.: Multistability of complex-valued neural networks with time-varying delays. Appl. Math. Comput. 294, 18–35 (2017)MathSciNetGoogle Scholar
  11. 11.
    Fang, T., Sun, J.T.: Further investigate the stability of complex-valued recurrent neural networks with time-delays. IEEE Trans. Neural Netw. Learn. Syst. 25, 1709–1713 (2014)CrossRefGoogle Scholar
  12. 12.
    Nitta, T.: Solving the XOR problem and the detection of symmetry using a single complex-valued neuron. Neural Netw. 16, 1101–1105 (2003)CrossRefGoogle Scholar
  13. 13.
    Pan, J., Liu, X.Z., Xie, W.C.: Exponential stability of a class of complex-valued neural networks with time-varying delays. Neurocomputing 164, 293–299 (2015)CrossRefGoogle Scholar
  14. 14.
    Song, Q.K., Yan, H., Zhao, Z.J., Liu, Y.R.: Global exponential stability of complex-valued neural networks with both time-varying delays and impulsive effects. Neural Netw. 79, 108–116 (2016)CrossRefGoogle Scholar
  15. 15.
    Velmurugan, G., Rakkiyappan, R., Cao, J.D.: Further analysis of global \(\mu \)-stability of complex-valued neural networks with unbounded time-varying delays. Neural Netw. 67, 14–27 (2015)CrossRefGoogle Scholar
  16. 16.
    Wang, Z.Y., Huang, L.H.: Global stability analysis for delayed complex-valued BAM neural networks. Neurocomputing 173, 2083–2089 (2016)CrossRefGoogle Scholar
  17. 17.
    Dupuis, P., Nagurney, A.: Dynamical systems and variational inequalities. Ann. Oper. Res. 44, 19–42 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Friesz, T.L., Bernstein, D.H., Mehta, N.J., Tobin, R.L., Ganjlizadeh, S.: Day-to-day dynamic network disequilibria and idealized traveler information systems. Oper. Res. 42, 1120–1136 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ding, K., Huang, N.J.: A new class of interval projection neural networks for solving interval quadratic program. Chaos Solitons Fractals 35, 718–725 (2008)CrossRefzbMATHGoogle Scholar
  20. 20.
    Friesz, T.L.: Dynamic Optimization and Differential Games. Springer, New York (2010)CrossRefzbMATHGoogle Scholar
  21. 21.
    Huang, B., Zhang, H.G., Gong, D.W., Wang, Z.S.: A new result for projection neural networks to solve linear variational inequalities and related optimization problems. Neural Comput. Appl. 23, 1753–1761 (2013)CrossRefGoogle Scholar
  22. 22.
    Nagumey, A., Zhang, D.: Projected Dynamical Systems and Variational Inequalities with Applications. Springer, New York (1996)Google Scholar
  23. 23.
    Xia, Y.S., Wang, J.: On the stability of global projected dynamical systems. J. Optim. Theory Appl. 106, 129–150 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Xia, Y.S., Wang, J.: A general projection neural network for solving monotone variational inequalities and related optimization problems. IEEE Trans Neural Netw. 15, 318–328 (2004)CrossRefGoogle Scholar
  25. 25.
    Zhang, S.C., Xia, Y.S., Wang, J.: A complex-valued projection neural network for constrained optimization of real functions in complex variables. IEEE Trans. Neural Netw. Learn. Syst. 26, 3227–3238 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Zhou, W., Song, Q.K.: Boundedness and complete stability of complex-valued neural networks with time delay. IEEE Trans. Neural Netw. Learn. Syst. 24, 1227–1238 (2013)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsSichuan UniversityChengduPeople’s Republic of China
  2. 2.College of Management ScienceChengdu University of TechnologyChengduPeople’s Republic of China

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