Model structure and boundary stabilization of an axially moving elastic tape

Linear and nonlinear equations of motion for a thin elastic tape moving axially between two sets of rollers with speed c > 0, and subject to a positive tension, were derived in an earlier article, and well-posedness was demonstrated in the linear case. Instability of the nominal equilibrium state for c sufficiently large was also demonstrated for the linear model, and it was shown that such instability can be overcome with active boundary control, exercised through movable rollers, synthesized via feedback on boundary data. In the present article we revisit this system, examining first of all some structural aspects for the fixed roller system, including the form of the adjoint system. In addition, we characterize a set of stabilizing boundary feedback controls for the movable roller configuration in a somewhat different manner than before, and we provide, for the corresponding closed loop systems, a simplified proof of well-posedness and a proof of uniform exponential stability.