On the asymptotic stability

in functional differential equations

Author:
A. O. Ignatyev

Journal:
Proc. Amer. Math. Soc. **127** (1999), 1753-1760

MSC (1991):
Primary 34K20

Published electronically:
February 11, 1999

MathSciNet review:
1636954

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Abstract | References | Similar Articles | Additional Information

Abstract: Consider a system of functional differential equations where is the vector-valued functional. The classical asymptotic stability result for such a system calls for a positive definite functional and negative definite functional . In applications one can construct a positive definite functional , whose derivative is not negative definite but is less than or equal to zero. Exactly for such cases J. Hale created the effective asymptotic stability criterion if the functional in functional differential equations is autonomous ( does not depend on ), and N. N. Krasovskii created such criterion for the case where the functional is periodic in . For the general case of the non-autonomous functional V. M. Matrosov proved that this criterion is not right even for ordinary differential equations. The goal of this paper is to prove this criterion for the case when is almost periodic in . This case is a particular case of the class of non-autonomous functionals.

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Additional Information

**A. O. Ignatyev**

Affiliation:
Institute for Applied Mathematics & Mechanics, R. Luxemburg Street, 74, Donetsk-340114, Ukraine

Email:
ignat@iamm.ac.donetsk.ua

DOI:
http://dx.doi.org/10.1090/S0002-9939-99-05094-7

Keywords:
Functional differential equations,
Lyapunov functionals,
asymptotic stability

Received by editor(s):
September 12, 1997

Published electronically:
February 11, 1999

Communicated by:
Hal L. Smith

Article copyright:
© Copyright 1999
American Mathematical Society