Asymptotic stability of the equilibrium of the damped oscillator

László Hatvani, Vilmos Totik, Roger Nussbaum

Research output: Contribution to journalArticlepeer-review

35 Scopus citations

Abstract

Conditions are given guaranteeing the property x(t) → 0, x(t) → 0 (t → ∞) for every solution of the equation x + h(t)x + k2x = 0 (t≥0, 0 < k =const.), where h is a nonnegative function. It is known that this property requires that in the average the damping coefficient h is not "too small" or "too large". In the first part we give a necessary and sufficient growth condition on h, provided that h is not "too small" in some integral sense. Then, considering the case of small h, we show that not only the size, but the distribution of the damping "bumps" is important. The main theorem takes into account both of them. Finally, we formulate theorems for the general case when h can be both small and large. It is pointed out that the conditions restricting h above and below are interdependent.

Original languageEnglish (US)
Pages (from-to)835-848
Number of pages14
JournalDifferential and Integral Equations
Volume6
Issue number4
StatePublished - Jul 1993

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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