Exponential stability and spectral analysis of the pendulum system under position and delayed position feedbacks

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Abstract
This article addresses the exponential stability and spectral analysis of the linearised pendulum system with both position and delayed position (PDP for short) feedbacks. The semigroup approach is adopted in investigation. The asymptotic spectral expression of the system is presented. It is shown that the spectrum of the system is located in the left half complex plane and its real part goes to −∞ at the high frequency end when the feedback gains and the delay satisfy some additional conditions. Therefore, exponential stability of the system follows from the spectrum determined growth property. Finally, some numerical simulations of a planar pendulum are presented.

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