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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Asymptotic stability in functional-differential equations by Liapunov functionals


Author: Bo Zhang
Journal: Trans. Amer. Math. Soc. 347 (1995), 1375-1382
MSC: Primary 34K20
DOI: https://doi.org/10.1090/S0002-9947-1995-1264834-9
MathSciNet review: 1264834
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Abstract: We consider the asymptotic stability in a system of functional differential equations $x’(t) = F(t,{x_t})$ by Liapunov functionals $V$. The work generalizes some well-known results in the literature in that we only require the derivative of $V$ to be negative definite on a sequence of intervals ${I_n} = [{S_n},{t_n}]$. We also show that it is not necessary to require a uniform upper bound on $V$ for nonuniform asymptotic stability.


References [Enhancements On Off] (What's this?)

    T. A. Burton, Uniform asymptotic stability in functional differential equations, Proc. Amer. Math. Soc. 68 (1978), 195-199. ---, Stability and periodic solutions of ordinary and functional differential equations, Academic Press, Orlando, 1985. T. A. Burton and L. Hatvani, Stability theorems for nonautonomous functional differential equations by Liapunov functionals, Tôhoku Math. J. 41 (1989), 65-104. ---, On nonuniform asymptotic stability for nonautonomous functional differential equations, Differential and Integral Equations 2 (1990), 285-293. T. A. Burton and G. Makay, Asymptotic stability for functional differential equations, Acta Math. Hungar. 65 (1994), 243-251. J. Hale, Theory of functional differential equations, Springer-Verlag, New York, 1977. J. Kato, A conjecture in Liapunov method for functional differential equations, Preprint. T. Yoshizawa, Stability by Liapunov’s second method, Math. Soc. Japan, Tokyo, 1966.

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Keywords: Functional differential equations, Liapunov functionals, asymptotic stability
Article copyright: © Copyright 1995 American Mathematical Society