On the asymptotic stability in functional differential equations
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- by A. O. Ignatyev PDF
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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.References
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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
- Received by editor(s): September 12, 1997
- Published electronically: February 11, 1999
- Communicated by: Hal L. Smith
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 1753-1760
- MSC (1991): Primary 34K20
- DOI: https://doi.org/10.1090/S0002-9939-99-05094-7
- MathSciNet review: 1636954