The Riesz basis property of a class of Euler-Bernoulli beam equation

In this paper, we prove that a sequence of generalized eigenvectors of a linear unbounded operator associated with an Euler-Bernoulli beam equation under bending moment boundary feedback forms a Riesz basis for the underlying state Hilbert space. As a consequence, the resulting closed-loop system is exponentially stable

Exponential Stabilization for Vibration Cable with a Tip Mass under Boundary Control and Non-Collocated Observation

We study the output feedback exponential stabilization for a vibration cable with tip mass. With only one measurement, we construct an infinite-dimensional state observer to trace the state and design an estimated state based controller to exponentially stabilize the original system. This is an essentially important improvement for the existence reference [O. Morgul, B.P. Rao and F. Conrad, IEEE Transactions on Automatic Control, 39(10) (1994), 2140-2145] where two measurements including the high order angular velocity feedback were adopted. When a control matched nonlinear internal uncertainty and external disturbance are taken into consideration, we construct an infinite-dimensional extended state observer (ESO) to estimate the total disturbance and state simultaneously. By compensating the total disturbance, an estimated state based controller is designed to exponentially stabilize the original system while making the closed-loop system bounded. Riesz basis approach is crucial to the verifications of the exponential stabilities of two coupled systems of the closed-loop system. Some numerical simulations are presented to illustrate the effectiveness.Comment: arXiv admin note: text overlap with arXiv:2011.0784