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Exponential Stability by the Linear Approximation

Exponential Stability by the Linear Approximation Di erential Equations, Vol. 37, No. 8, 2001, pp. 1057{1073. Translated from Di erentsial'nye Uravneniya, Vol. 37, No. 8, 2001, pp. 1011{1027. Original Russian Text Copyright c 2001 by Izobov. SURVEY ARTICLES N. A. Izobov Institute for Mathematics, National Academy of Sciences, Minsk, Belarus Received January 18, 2001 INTRODUCTION The rst Lyapunov method for studying the exponential stability of di erential systems by the linear approximation is based on the notion of the Lyapunov exponent ([1, p. 27]; see also [2, p. 17]) [f ]  lim t lnkf (t)k of a piecewise continuous vector or matrix function f on the half-line t  0. t!+1 Notation. Statement of the Problems We consider n-dimensional real di erential systems. Namely, we deal with a linear rst-approx- imation system x _ = A(t)x; x 2 R;n  2;t  0; (1 ) with piecewise continuous (actually, measurability is sucient) bounded [kA(t)k a< +1 for t  0] coecients and perturbed nonlinear systems y _ = A(t)y + f (t;y);y 2 R;n  2;t  0; (2) where the vector functions f (t;y) are piecewise continuous in t  0 and continuous in y 2 U fy 2 R : kyk <% http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

Exponential Stability by the Linear Approximation

Izobov, N.
Differential Equations , Volume 37 (8) – Oct 12, 2004
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Publisher
Springer Journals
Copyright
Copyright © 2001 by MAIK “Nauka/Interperiodica”
Subject
Mathematics; Difference and Functional Equations; Ordinary Differential Equations; Partial Differential Equations
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1023/A:1012416516936
Publisher site
See Article on Publisher Site

Abstract

Di erential Equations, Vol. 37, No. 8, 2001, pp. 1057{1073. Translated from Di erentsial'nye Uravneniya, Vol. 37, No. 8, 2001, pp. 1011{1027. Original Russian Text Copyright c 2001 by Izobov. SURVEY ARTICLES N. A. Izobov Institute for Mathematics, National Academy of Sciences, Minsk, Belarus Received January 18, 2001 INTRODUCTION The rst Lyapunov method for studying the exponential stability of di erential systems by the linear approximation is based on the notion of the Lyapunov exponent ([1, p. 27]; see also [2, p. 17]) [f ]  lim t lnkf (t)k of a piecewise continuous vector or matrix function f on the half-line t  0. t!+1 Notation. Statement of the Problems We consider n-dimensional real di erential systems. Namely, we deal with a linear rst-approx- imation system x _ = A(t)x; x 2 R;n  2;t  0; (1 ) with piecewise continuous (actually, measurability is sucient) bounded [kA(t)k a< +1 for t  0] coecients and perturbed nonlinear systems y _ = A(t)y + f (t;y);y 2 R;n  2;t  0; (2) where the vector functions f (t;y) are piecewise continuous in t  0 and continuous in y 2 U fy 2 R : kyk <%

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Differential Equations – Springer Journals

Published: Oct 12, 2004

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