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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.
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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

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