Integral conditions on the asymptotic stability for the damped linear oscillator with small damping.

*(English)*Zbl 0844.34051The 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 |

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\textit{L. Hatvani}, Proc. Am. Math. Soc. 124, No. 2, 415--422 (1996; Zbl 0844.34051)

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