Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-14T05:08:01.192Z Has data issue: false hasContentIssue false
  • English
  • Français

A Liapunov functional for a scalar differential difference equation

Published online by Cambridge University Press:  14 November 2011

E. F. Infante  and
J. A. Walker
E. F. Infante
Affiliation:
Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, R.I., U.S.A.
J. A. Walker
Affiliation:
The Technological Institute, Northwestern University, Evanston, Illinois, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

Given the scalar, retarded differential difference equation x'(t)=ax(t) +bx(t−τ), a quadratic functional in explicit form is obtained that yields necessary and sufficient conditions for the asymptotic stability of this equation. This functional a Liapunov functional, is obtained through the study of the Liapunov functions associated with a difference equation approximation of the difference differential equation. The functional then obtained not only yields necessary and sufficient conditions for asymptotic stability, but provides estimates for rates of decay of the solutions as well as conditions, for asymptotic stability independent of the magnitude of the delay τ.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 79 , Issue 3-4 , 1978 , pp. 307 - 316
Copyright
Copyright © Royal Society of Edinburgh 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Banks, H. T. and Burns, J. A. An Abstract Framework for Approximate Solutions to Optimal Control Problems Governed by Hereditary Systems. Proc. Internal. Conference Differential Equations, 10–25 (London: Academic Press, 1975).CrossRefGoogle Scholar
2Datko, R. An Algorithm for Computing Liapunov Functionals for Some Differential Difference Equations. Ordinary Differential Equations. Proc. 1971 Naval Res. Lab. Mat. Res. Cen. Conference, 387–397 (London: Academic Press, 1972).Google Scholar
3Hale, J. K. Functional Differential Equations. Appl. Math. Sci. Ser. (Berlin: Springer, 1971).Google Scholar
4Hale, J. K. Theory of Functional Differential Equations. Appl. Math. Sci. Ser. (Berlin: Springer, 1977).Google Scholar
5LaSalle, J. P. An Invariance Principle in the Theory of Stability. In Differential Equations and Dynamical Systems, 277–286 (London: Academic Press, 1967).Google Scholar
6LaSalle, J. P., The Stability and Control of Discrete Processes. Appl. Math. Sci. Ser. (Berlin: Springer, to appear).Google Scholar
7Repin, I. M.Quadratic Liapunov Functionals for Systems with Delays. Prikl. Mat. Meh. 29 (1965), 564–566.Google Scholar
8Walker, J. A.On the Application of Liapunov's Direct Method to Linear Dynamical Systems. J. Math. Anal. Appl. 53 (1976), 187–220.CrossRefGoogle Scholar
9Zivotovskii, L. A.Absolute Stability of the Solutions of Differential Equations with Retarded Arguments. Trudy Sent. Teor. Differencial Uravneniis Otklon. Argumentom Univ. Družby Narodov Patrisa Lumumby 7 (1969), 82–91.Google Scholar