Boundary stabilization for axially moving Kirchhoff string under fractional PI control

Abstract

AbstractIn this paper, we investigate vibration control for an axially moving Kirchhoff string. A fractional PI boundary control is applied transversally to the right boundary and the transverse vibration of the system is shown to be suppressed. By constructing an auxiliary equivalent system (PDE‐ODE) and a novel energy like function, the well‐posedness and exponential stability of the closed‐loop system are established by means of the Faedo‐Galerkin method and the Lyapunov method. The effectiveness of the proposed controller is verified by a numerical simulation using finite element method.