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2006 On the partial asymptotic stability in nonautonomous differential equations
Oleksiy Ignatyev
Differential Integral Equations 19(7): 831-839 (2006).

Abstract

A system of ordinary differential equations $dx/dt=X(t,x)$ which has a zero solution $x=0$ is considered. It is assumed that there exists a function $V(t,x)$, positive definite with respect to part of state variables such that its derivative $dV/dt$ is nonpositive. It is proved that if the function $\sum_{i=1}^jV_i^2$ is positive definite with respect to part of the studying variables, then the zero solution is asymptotically stable with respect to these variables. Here $V_1=dV/dt, V_{i}=dV_{i-1}/dt, \quad i=2, \dots,j;\quad j$ is some positive integer. The instability criterion is also obtained.

Citation

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Oleksiy Ignatyev. "On the partial asymptotic stability in nonautonomous differential equations." Differential Integral Equations 19 (7) 831 - 839, 2006.

Information

Published: 2006
First available in Project Euclid: 21 December 2012

zbMATH: 1212.34154
MathSciNet: MR2235897

Subjects:
Primary: 34D20

Rights: Copyright © 2006 Khayyam Publishing, Inc.

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Vol.19 • No. 7 • 2006
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