Stabilization of a pendulum in dynamic boundary feedback with a memory type heat equation

This paper addresses a dynamic feedback stabilization of an interconnected pendulum system with a memory type heat equation, where the kernel memory is an exponential polynomial. By introducing some new variables, the time-variant system is transformed into a time-invariant one. The detailed spectral analysis is presented. Remarkably, the resolvent of the closed-loop system operator is not compact anymore. The residual spectrum is shown to be empty and the continuous spectrum consists of finite isolated points. Furthermore, it is shown that there is a sequence of generalized eigenfunctions, which forms a Riesz basis for the Hilbert state space. This deduces the spectrum-determined growth condition for the C0-semigroup, and the exponential stability is then followed. Finally, some numerical simulations are presented to show the effectiveness of this feedback control design.