Abstract
In this paper, a slight generalization of the celebrated continuous Halanay inequality is developed, a discrete analogue of the continuous Halanay inequality is proved. Using continuous and discrete Halanay time-delayed inequalities, we derived some sufficient conditions for the global exponential (asymptotic) stability of equilibrium point of neural networks.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Halanay, A.: Differential Equations: Stability, Oscillations, Time Lags. Academic Press, New York (1966)
Gopalsamy, K.: Stability and Oscillations in Delay Differential Equations of Population Dynamics. Springer, Cambridge (1992)
Liz, E., Trofimchuk, S.: Existence and Stability of Almost Periodic Solutions for Quasilinear Delay Systems and Halanay Inequality. J. Math. Anal. Appl. 248, 625–644 (2000)
Mohamad, S., Gopalsamy, K.: Continuous and Discrete Halanay-type Inequalities. Bull. Aus. Math. Sot. 61, 371–385 (2000)
Pinto, M., Trofimchu, S.: Stability and Existence of Multiple Periodic Solutions for A Quasilinear Differential Equation with Maxima. Proceedings of the Royal Society of Edinburgh: Section A Mathematics 130, 1103–1118 (2000)
Cooke, K.L., Ivanov, A.F.: On the Discretization of A Delay Differential Equation. J. Differ. Equations Appl. 6, 105–119 (2000)
Andreev, A.: On the Stability of Nonautonomous Functional Differential Equations. Nonlinear Anal. 30, 448–457 (1997)
Cermák, J.: The Asymptotic Bounds of Solutions of Linear Delay Systems. Journal of Mathematical Analysis and Applications 2, 373–388 (1998)
Hatvani, L.: On Lyapunov,s Direct Method for Nonautonomous FDE,s. Functional Differential equations 5, 315–323 (1998)
Hatvani, L.: On the Asymptotic Stability for Functional Differential Equations by Lyapunov Functionals. Nonlinear Anal. 40, 251–263 (2000)
Lipovan, O.: A retarded Gronwall-like Inequality and Its Applications. Joural of Mathematical Analysis and Applications 252, 389–401 (2000)
Zeng, Z.G., Wang, J., Liao, X.X.: Global Exponential Stability of A General Class of Recurrent Neural Networks with Time-varying Delays. IEEE Trans. Circuits and Systems 50, 1353–1358 (2003)
Liz, E., Ferreiro, J.B.: A Note on the Global Stability of Generalized Difference Equations. Applied Mathematics Letters 15, 655–659 (2002)
Zeng, Z.G., Wang, J., Liao, X.X.: Stability Analysis of Delayed Cellular Neural Networks Described Using Cloning Templates. IEEE Trans. Circuits and Syst. 51, 2313–2324 (2004)
Zeng, Z.G., Wang, J.: Global Exponential Stability of Recurrent Neural Networks with Time-varying Delays in the Presence of Strong External Stimuli. Neural Networks 19, 1528–1537 (2006)
Zeng, Z.G., Wang, J.: Multiperiodicity and Exponential Attractivity Evoked by Periodic External Inputs in Delayed Cellular Neural Networks. Neural Computation 18, 848–870 (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Huang, J., Liu, J. (2009). Continuous and Discrete Halanay Delayed Inequalities and Their Applications in Stability of Neural Networks. In: Yu, W., He, H., Zhang, N. (eds) Advances in Neural Networks – ISNN 2009. ISNN 2009. Lecture Notes in Computer Science, vol 5551. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01507-6_31
Download citation
DOI: https://doi.org/10.1007/978-3-642-01507-6_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-01506-9
Online ISBN: 978-3-642-01507-6
eBook Packages: Computer ScienceComputer Science (R0)