Dynamic stabilization of an Euler–Bernoulli beam under boundary control and non-collocated observation

We study the dynamic stabilization of an Euler–Bernoulli beam system using boundary force control at the free end and bending strain observation at the clamped end. We construct an infinite-dimensional observer to track the state exponentially. A proportional output feedback control based on the estimated state is designed. The closed-loop system is shown to be non-dissipative but admits a set of generalized eigenfunctions, which forms a Riesz basis for the state space. As consequences, both the spectrum-determined growth condition and exponential stability are concluded.