Stability of an Euler-Bernoulli Beam With a Nonlinear Dynamic Feedback System

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 finite-dimensional nonlinear dynamic feedback system. The latter comprises dynamic systems satisfying the nonlinear KYP lemma, which may represent the closed-loop dynamics of subordinate controlled actuators, 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 partial differential equation–ordinary differential equation (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.