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.

**1.**M. S. Berger and Y. Y. Chen,*Forced quasiperiodic and almost periodic solution for nonlinear systems*, Nonlinear Anal.**21**(1993), no. 12, 949–965. MR**1249212**, 10.1016/0362-546X(93)90118-C**2.**A. S. Besicovitch,*Almost periodic functions*, Dover Publications, Inc., New York, 1955. MR**0068029****3.**Harald Bohr,*Almost Periodic Functions*, Chelsea Publishing Company, New York, N.Y., 1947. MR**0020163****4.**Sonoko Mori,*On 𝑆5 type modal logics based on intermediate logics with finitely many valued linear models*, J. Tsuda College**10**(1978), 35–43. MR**0479948****5.**C. Corduneanu,*Almost Periodic Functions*, 2nd edition, Chelsea Publ. Co., New York, 1989.**6.**A. M. Fink,*Almost periodic differential equations*, Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974. MR**0460799****7.**Wolfgang Hahn,*Stability of motion*, Translated from the German manuscript by Arne P. Baartz. Die Grundlehren der mathematischen Wissenschaften, Band 138, Springer-Verlag New York, Inc., New York, 1967. MR**0223668****8.**Jack Hale,*Theory of functional differential equations*, 2nd ed., Springer-Verlag, New York-Heidelberg, 1977. Applied Mathematical Sciences, Vol. 3. MR**0508721****9.**V. B. Kolmanovskiĭ and V. R. Nosov,*Stability of functional-differential equations*, Mathematics in Science and Engineering, vol. 180, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1986. MR**860947****10.**N. N. Krasovskiĭ,*Stability of motion. Applications of Lyapunov’s second method to differential systems and equations with delay*, Translated by J. L. Brenner, Stanford University Press, Stanford, Calif., 1963. MR**0147744****11.**B. M. Levitan and V. V. Zhikov,*Almost periodic functions and differential equations*, Cambridge University Press, Cambridge-New York, 1982. Translated from the Russian by L. W. Longdon. MR**690064****12.**V. M. Matrosov,*On the theory of stability of motion*, J. Appl. Math. Mech.**26**(1962), 1506–1522. MR**0150404****13.**Nicolas Rouche, P. Habets, and M. Laloy,*Stability theory by Liapunov’s direct method*, Springer-Verlag, New York-Heidelberg, 1977. Applied Mathematical Sciences, Vol. 22. MR**0450715****14.**George Seifert,*On uniformly almost periodic sets of functions for almost periodic differential equations*, Tôhoku Math. J. (2)**34**(1982), no. 2, 301–309. MR**664736**, 10.2748/tmj/1178229256**15.**T. Yoshizawa,*Stability theory and the existence of periodic solutions and almost periodic solutions*, Springer-Verlag, New York-Heidelberg, 1975. Applied Mathematical Sciences, Vol. 14. MR**0466797**

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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:
https://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