Stabilization analysis of a generalized nonlinear axially moving string by boundary velocity feedback

This paper addresses vibration suppression of an axially moving nonlinear string subject to boundary control at either the downstream or the upstream end. The equation of motion and the boundary conditions are derived from Hamilton's Principle, using the exact expression of the strain to describe the geometrical nonlinearity due to the finite transverse deformation. We can approximate this model by various simpler models found in the literature. We show that the oscillations of the moving string can be stabilized exponentially via linear negative velocity feedback acting at one end of the string. Our proofs rely on Lyapunov's direct method, using a carefully constructed energy functional.