Abstract
The asymptotic stability of delay differential equation x′ (t) = Ax (t) + Bx (t τ) is concerned with, where A, B ε C d×d are constant complex matrices, x (t τ) = (x 1(t − τ 1),x 2(t − τ 2,...,x d (t − τ d ))T, τ k > 0 (k = 1,...,d) stand for constant delays. Two criteria through evaluation of a harmonic function on the boundary of a certain region are obtained. The similar results for neutral delay differential equation x′(t)=Lx(t)+Mx(t−τ)+Nx′(t−τ) are also obtained, where L, M and N ε C d×d are constant complex matrices and τ>0 stands for constant delay. Numerical examples are showed to check the results which are more general than those already reported.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Hu Guangda, Stability and numerical analysis for delay differential equations (to appear).
Hu Guangda, Hu Guangdi, Some simple criteria for the stability of NDDEs, Appl. Math. Comput., 1996,80(2–3):257–271.
Hu Guangda, Hu Guangdi, Stability of neutral delay-differential system: boundary criteria, Appl. Math. Comput., 1997,87:247–259.
Kuang, J. X., Xiang, J. X., Tian, H. J., The asymptotic stability of one-parameter methods for neutral differential equations, BIT, 1994,34:400–408.
Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, Boston: Kluwer Academic Publishers, 1992.
Hale, J. K., Infante, E. F., Tsen, F. S. P., Stability in linear delay equations, J. Math. Anal. Appl., 1985,105:533–555.
Brayton, R. K., Willoughby, R. A., On the numerical integration of a symmetric system of difference-differential equation of neutral type, J. Math. Anal. Appl., 1967,18:182–189.
Li, L. M., Stability of linear neutral delay-differential system, Bull. Austral Math. Soc., 1988,38:339–344.
Sun, L. P., Stability of neutral delay differential equations: boundary criteria. J. Shanghai Teachers University, 2001,30(4):34–39.
Khusaina, D. Ya., Yunkova, E. A., Investigation of the stability of linear systems of neutral type by the Lyapunov function method, Diff. Uravn, 1988,24:613–621.
Kolmanovskii, V., Myshkis, A., Applied Theory of Functional Differential Equations, Dordrecht: Kluwer Academic Publishers, 1992.
Hu Guangdi, Hu Guangda, Stability of discrete-delay system: boundary criteria, Applied Mathematics and Computation, 1996,80:95–104.
Hu Guangda, Taketomo Mitsui, Stability of linear delay differential systems with matrices having common eigenvectors, Japan J. Indust. Appl. Math., 1996,13:487–494.
Lu, L. H., Numerical stability of θ-methods for systems of differential equations with several delays, J. CAM, 1991,34:291–304.
Kuang, J. X., Theory of Numerical Analysis for Functional Differential Equations, Beijing: Science Press, 1999.
Ioakimidis, N. I., Anastasselou, E. G., A new, simple approach to the derivation of exact analytical formulae for the zeros of analytic functions, Appl. Math. Comput., 1985,17(2):123–127.
Hu Guangda, Hu Guangdi, A relation between the weighted logarithmic norm of a matrix and the Lyapunov equation, BIT, 2000,40(3):606–610.
Dekker, K., Verwer, J. G., Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, Amsterdam: North-Holland, 1984.
Desoer, C. A., Vidyasagar, M., Feedback System: Input-Output Properties, New York: Academic Press, 1977.
Driver, R. D., Ordinary and delay differential equations, New York: Springer-Verlag, 1977.
Lancaster Peter, Tismenetask, M., The theory of matrices, New York: Academic Press, 1985.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sun, L. Stability analysis for linear delay differential equations and numerical examples. Appl. Math. Chin. Univ. 18, 390–402 (2003). https://doi.org/10.1007/s11766-003-0066-6
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11766-003-0066-6