Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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: Consider a system of functional differential equations $dx/dt=f(t,x_{t})$ where $f$ is the vector-valued functional. The classical asymptotic stability result for such a system calls for a positive definite functional $V(t,\varphi )$ and negative definite functional ${dV}/{dt}$. In applications one can construct a positive definite functional $V$, 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 $f$ in functional differential equations is autonomous ($f$ does not depend on $t$), and N. N. Krasovskii created such criterion for the case where the functional $f$ is periodic in $t$. For the general case of the non-autonomous functional $f$ 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 $f$ is almost periodic in $t$. 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