Stabilization Of An Euler-Bernoulli Beam With Disturbances In Dynamic Boundaries

With extensive applications of elasticity structures,more and more scholars devoted to study the stabilization of the system of elasticity with boundary disturbances.In this paper,under two different dynamic boundary conditions of the following Euler-Bernoulli beam systems with disturbances,feedback controllers are designed,respectively.Moreover,the well-posedness and the stability are analyzed:(1)Constrained one-link Euler-Bernoulli beam system which is linked to a motor at one end,while the antenna at the other end is constrained by the contact force.The disturbance is canceled via the controller at the motor.(2)Rigid-flexible hybrid Euler-Bernoulli beam which is clamped at one end,and is fastened at the other end to a flexible rigid antenna,where elastic vibrations are presented by the Euler-Bernoulli beam equation,while the oscillations of the antenna with disturbances are described by Newton-Euler equations.The controllers are assembled at the flexible rigid antenna so as to stabilized the system.The active disturbance rejection control approach is adopted in investigation.Time varying high gain extended state observers are first designed to estimate disturbances,and then boundary feedback controllers are proposed to cancel the disturbances.Moreover,the well-posedness of the closed-loop system is gained by the semigroup theory.Furthermore,under different assumptions of the time varying high gain functions,the asymptotical stability and the exponential stability of the closed-loop system are proved via the Lyapunov function approach.Finally,numerical simulations are presented to validate the theoretical conclusions.