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Integral conditions on the asymptotic stability for the damped linear oscillator with small damping. (English) Zbl 0844.34051
The author considers the equation (1) $$x'' + h(t)x' + k^2x = 0$$ under the assumption (2) $$0 \leq h(t) \leq \overline h < \infty$$. He proves that the condition $$\limsup_{t \to \infty} (t^{- 2/3} \int^t_0 h(s) ds) > 0$$ is sufficient for the asymptotic stability of $$x = x' = 0$$, and the exponent $$2/3$$ is best possible. He obtains this as a corollary of a general result on intermittent damping which reads as follows: Suppose that (2) is satisfied. If there exists a sequence $$\{I_n\}$$ of non-overlapping intervals such that $\sum^\infty_{n = 1} {1 \over 1 + |I_n |^2} \left( \int^t_0 h(s)ds \right)^3 = \infty,$ then the zero solution of (1) is asymptotically stable. Moreover, the exponent 3 in the statement is best possible.
Reviewer: W.Müller (Berlin)

##### MSC:
 34D20 Stability of solutions to ordinary differential equations 34A30 Linear ordinary differential equations and systems
##### Keywords:
asymptotic stability; intermittent damping
Full Text:
##### References:
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