×

Exponential stability of continuous-time and discrete-time cellular neural networks with delays. (English) Zbl 1030.34072

Summary: Convergence characteristics of continuous-time cellular neural networks with discrete delays are studied. By using Lyapunov functionals, we obtain delay independent sufficient condition for the networks to converge exponentially toward the equilibria associated with the constant input sources. Halanay-type inequalities are employed to obtain sufficient conditions for the networks to be globally exponentially stable. It is shown that the estimates obtained from the Halanay-type inequalities improve the estimates obtained from the Lyapunov methods. Discrete-time analogues of the continuous-time cellular neural networks are formulated and studied. It is shown that the convergence characteristics of the continuous-time systems are preserved by the discrete-time analogues without any restriction imposed on the uniform discretization step size.

MSC:

34K20 Stability theory of functional-differential equations
92B20 Neural networks for/in biological studies, artificial life and related topics
34K60 Qualitative investigation and simulation of models involving functional-differential equations
39A99 Difference equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] C.T.H. Baker, A. Tang, Generalised Halanay inequalities for Volterra functional differential equations and discretised versions, Report no. 299, Machester Centre for Computational Mathematics, The University of Manchester, England, 1996
[2] ()
[3] Cao, J., Global stability analysis in delayed cellular neural networks, Phys. rev. E, 59, 5940-5944, (1999)
[4] Cao, J., Periodic solutions and exponential stability in delayed cellular neural networks, Phys. rev. E, 60, 3244-3248, (1999)
[5] Cao, J.; Zhou, D., Stability analysis of delayed cellular neural networks, Neural networks, 11, 1601-1605, (1998)
[6] Chua, L.O.; Yang, L., Cellular neural networks: theory, IEEE trans. circuits syst., 35, 1257-1272, (1988) · Zbl 0663.94022
[7] Chua, L.O.; Yang, L., Cellular neural networks: applications, IEEE trans. circuits syst., 35, 1273-1290, (1988)
[8] Chua, L.O.; Roska, T., Stability of a class of nonreciprocal cellular neural networks, IEEE trans. circuits syst., 37, 1520-1527, (1990)
[9] Civalleri, P.P.; Gilli, M., On stability of cellular neural networks with delay, IEEE trans. circuits syst., 40, 157-165, (1993) · Zbl 0792.68115
[10] Devaney, R.L., An introduction to chaotic dynamical systems, (1989), Addison-Wesley California · Zbl 0695.58002
[11] Gilli, M., Stability of cellular neural networks and delayed cellular neural networks with nonpositive templates and nonmonotonic output functions, IEEE trans. circuits syst., 41, 518-528, (1994)
[12] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Publishers The Netherlands · Zbl 0752.34039
[13] Guzelis, C.; Chua, L.O., Stability analysis of generalised cellular neural networks, Int. J. circuit theor. appl., 21, 1-33, (1993) · Zbl 0766.94024
[14] Halanay, A., Differential equations, (1966), Academic Press New York
[15] Iserles, A., Stability and dynamics of numerical methods for nonlinear ordinary differential equations, IMA J. numer. anal., 10, 1-30, (1990) · Zbl 0686.65054
[16] Istratescu, V.I., Fixed point theory D, (1981), Riedel Dordrecht
[17] Liu, D.; Michel, A.N., Cellular neural networks for associative memories, IEEE trans. circuits syst., 40, 119-121, (1993) · Zbl 0800.92046
[18] Matsumoto, T.; Chua, L.O.; Suzuki, H., CNN cloning template: connected component detector, IEEE trans. circuits syst., 37, 633-635, (1990) · Zbl 0964.94501
[19] Mickens, R.E., Nonstandard finite difference models of differential equations, (1994), World Scientific Singapore · Zbl 0925.70016
[20] Mohamad, S.; Gopalsamy, K., Continuous and discrete halanay-type inequalities, Bull. aust. math. soc., 61, 371-385, (2000) · Zbl 0958.34008
[21] Prufer, M., Turbulence in multistep methods for initial value problems, SIAM J. appl. math., 45, 32-69, (1985) · Zbl 0576.65067
[22] Roska, T.; Chua, L.O., Cellular neural networks with nonlinear and delay-type template, Int. J. circuit theor. appl., 20, 469-481, (1992) · Zbl 0775.92011
[23] Stewart, I., Warning-handle with care, Nature, 355, 16-17, (1992)
[24] Stuart, A.M.; Humphries, A.R., Dynamical systems and numerical analysis, (1996), Cambridge University Press Cambridge · Zbl 0869.65043
[25] Takahashi, N.; Chua, L.O., A new sufficient condition for nonsymmetric CNN’s to have a stable equilibrium point, IEEE trans. circuits syst., 44, 1092-1095, (1997)
[26] Takahashi, N.; Chua, L.O., On the complete stability of nonsymmetric cellular neural networks, IEEE trans. circuits syst., 45, 754-758, (1998) · Zbl 0952.94023
[27] Ushiki, S., Central difference scheme and chaos, Phys. D, 4, 407-424, (1982) · Zbl 1194.65097
[28] Yee, H.C.; Sweby, P.K.; Griffiths, D.F., A study of spurious asymptotic numerical solutions of nonlinear differential equations by the nonlinear dynamics approach, (), 259-267 · Zbl 0760.65087
[29] Yee, H.C.; Sweby, P.K.; Griffiths, D.F., Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. I. the dynamics of time discretization and its implications for algorithm development in computational fluid dynamics, J. comput. phys., 97, 249-310, (1991) · Zbl 0760.65087
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.