Uniform asymptotic stability in functional differential equations
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- by T. A. Burton
- Proc. Amer. Math. Soc. 68 (1978), 195-199
- DOI: https://doi.org/10.1090/S0002-9939-1978-0481371-5
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Abstract:
The classical uniform asymptotic stability result for a system of functional differential equations \begin{equation}\tag {$1$} x’ = F(t,{x_t})\end{equation} calls for a Liapunov functional $V(t,\phi )$ satisfying $W(|\phi (0)|) \leqslant V(t,\phi ) \leqslant {W_1}(|\phi (0)|) + {W_2}(|||\phi |||),{V’_{(1)}} \leqslant - {W_3}(|\phi (0)|)$, and $|f(t,{x_t})|$ bounded for $|||{x_t}|||$ bounded. We show that it is not necessary to require $|f(t,{x_t})|$ bounded. Here, $||| \cdot |||$ is the ${L^2}$-norm.References
- Taro Yoshizawa, Stability theory by Liapunov’s second method, Publications of the Mathematical Society of Japan, vol. 9, Mathematical Society of Japan, Tokyo, 1966. MR 0208086
Bibliographic Information
- © Copyright 1978 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 68 (1978), 195-199
- MSC: Primary 34K20
- DOI: https://doi.org/10.1090/S0002-9939-1978-0481371-5
- MathSciNet review: 0481371