Stability of an Euler-Bernoulli beam with a nonlinear dynamic feedback system

TL;DR

This paper rigorously analyzes the stability of a lossless Euler-Bernoulli beam with a nonlinear dynamic feedback system at its tip, proving global well-posedness and asymptotic stability using semigroup theory.

Contribution

It introduces a novel stability analysis for a beam with nonlinear boundary controllers and passive environment, establishing global stability results for the coupled PDE-ODE system.

Findings

Proves global-in-time well-posedness of the system

Establishes asymptotic stability of the closed-loop system

Derives uniform bounds on solutions and derivatives

Abstract

This paper is concerned with the stability analysis of a lossless Euler-Bernoulli beam that carries a tip payload which is coupled to a nonlinear dynamic feedback system. This setup comprises nonlinear dynamic boundary controllers satisfying the nonlinear KYP lemma as well as the interaction with a nonlinear passive environment. Global-in-time wellposedness and asymptotic stability is rigorously proven for the resulting closed-loop PDE-ODE system. The analysis is based on semigroup theory for the corresponding first order evolution problem. For the large-time analysis, precompactness of the trajectories is shown by deriving uniform-in-time bounds on the solution and its time derivatives.