Abstract
In this paper, a class of delay differential equations with nonlinear impulsive control is discussed. Based on the nonsmooth analysis, criteria of stability are obtained for delay differential equations with nonlinear impulses control under certain conditions. These criteria can be applied to some neural network models. At the end of the paper, two examples are provided to illustrate the feasibility and effectiveness of the proposed results.
Similar content being viewed by others
References
T. Erneux, Applied Delay Differential Equations, Springer, Berlin, 2009.
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Volume 191 in the series of Mathematics in Science and Engineering, Academic Press, New York, 1993.
J. Wiener and J. K. Hale, Ordinary and Delay Differential Equations, J. Wiley, New York, 1992.
K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population, Kluwer Academic Press, Dordrecht, 1992.
J. Kuang and Y. H. Cong, Stability of Numerical Methods for Delay Differential Equations, Science Press, Beijing, 2005.
M. Forti, On global asymptotic stability of a class of nonlinear system arising in neural networks theory, J. Diff. Equs., 1994, 113: 246–264.
J. D. Cao and L. Wang, Periodic oscillatory solution of bidirectional associative memory networks with delays, Phys. Rev., 2000, 61: 1825–1828.
S. J. Guo and L. H. Huang, Periodic oscillation for a class of neural networks with variable coefficients, Nonlin. Anal., 2004, 6: 545–561.
H. T. Lu, Global exponential stability analysis of Cohen-Grossberg neural networks, IEEE Trans. Circuits & Systems II: Express Briefs, 2005, 52: 476–479.
H. Akca, R. Alassar, and V. Corachav, Continuous-time additive Hopfield-type networks with impulses, J. Math. Anal. Appl., 2004, 290: 436–451.
S. Mohamad and K. Gopalsamy, Exponential stability preservation in semi-discretisations of networks with nonlinear impulses, Commun. Nonlin. Sci. Numer. Simul., 2009, 14: 27–50.
P. van den Driessche and X. Zou, Global atractivity in delayed Hopfield neural networks models, SIAM. J. Appl. Math., 1998, 58: 1878–1890.
Z. K. Huang and Y. H. Xia, Exponential p-stability of second order Cohen-Grossberg neural networks with transmission delays and learning behavior, Simul. Modelling Prac. Theory, 2007, 15: 622–634.
Y. Zhao, Q. S. Lu, and Z. Feng, Stability for the mix-delayed Cohen-Grossberg neural networks with nonlinear impulse, Journal of Systems Science & Complexity, 2010, 23(3): 665–680.
K. Yuan and J. D. Cao, An analysis of global asymptotic stability of delayed Cohen-Grossberg neural networks via nonsmooth analysis, IEEE Trans. Circuits Syst., 2005, 345: 1854–1861.
Z. Wen and J. T. Sun, Global Asymptotic stability of delayed BAM neural networks with impulses via nonsmooth analysis, Neurocomputing, 2008, 71: 1543–1549.
H. Guan, D. J. Hill and X. M. Shen, On hybrid impulsive and switching systems and application to nonlinear control, IEEE Trans. Auto. Control, 2005, 50: 1058–1062
C. X. Li and J. T. Sun, Stability analysis of nonlinear stochastic differential delay systems under impulsive control, Phys. Lett., 2010, 374: 1154–1158.
S. Mohamad and K. Gopalsamy, A unified treatment for stability preservation in computer simulations of impulsive BAM networks, Comput. Math. Appl., 2008, 55: 2043–2063.
Z. K. Huang and Y. H. Xia, Global exponential stability of BAM neural networks with transmission delays and nonlinear impulsive, Chaos, Solitons and Fractals, 2008, 38: 489–498.
R. T. Rockafellar and R. J. B. Wets, Variational Analysis, Springer-Verlag, Berlin/Heideberg, 1998.
F. H. Clarke, Optimization a Nonsmooth Analysis, Wiley, New York, 1983.
B. H. Pourciau, Hadamard theorem for Lacally Lipschitzian maps, J. Math. Anal. Appl., 1982, 85: 279–285.
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Science, Academic Press, New York, 1994.
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper is supported by Natural Science Foundation of China under Grant Nos. 10972018 and 11072013.
This paper was recommended for publication by Editor Bingyu ZHANG.
Rights and permissions
About this article
Cite this article
Zhao, Y., Lu, Q., Feng, Z. et al. Delay differential equations under nonlinear impulsive control and applications to neural network models. J Syst Sci Complex 25, 707–719 (2012). https://doi.org/10.1007/s11424-012-1110-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-012-1110-5