On the control of axially moving material systems

Vibration control can improve the performance of many axially moving material systems (e.g., web handling machines and tape drives). Researchers have used Lyapunov analysis to develop vibration stabilizing controllers for distributed parameter models of axially moving material systems. Both the material and regular time derivatives have been used in these analyses despite the fact that they give different results. This paper proves that for a pinned axially moving string model: (i) Lyapunov stability analysis using the material derivative incorrectly predicts that a time-varying functional is constant and (ii) neglect of the coupled domain in boundary control analysis is ill posed and incorrectly predicts bounded forced response and exponentially decaying transients.