Abstract
A damped harmonic oscillator x ̈ ( t ) + a 1 x ̇ ( t ) + a 0 x ( t ) = 0 with a 0 , a 1 > 0 is known to be exponentially stable. We extend this result to time-varying positive coefficients a 0 ( t ) , a 1 ( t ) , t ≥ 0 , which are bounded from above and below and satisfy sup t ≥ 0 a 0 ( t ) < ( inf t ≥ 0 a 1 ( t ) ) 2 and we thus further extend the sufficient condition sup t ≥ 0 a 0 ( t ) ≤ 1 4 ( inf t ≥ 0 a 1 ( t ) ) 2 by Levin (1969). Under slightly weaker assumptions we show uniform stability.
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