Boundary Stabilization for a Class of Nonlinear Axially Moving Systems With Coupling Transverse and Longitudinal Vibrations

This paper discusses boundary stabilization within the framework of absolute stability for axially moving systems experiencing transverse and longitudinal coupling vibrations. These vibrations are described by a nonlinear coupled beam model derived from Hamilton's principle. A boundary feedback sector criterion is proposed, encompassing a wide array of linear and nonlinear functions, to reduce vibrations within the axially moving systems by measuring boundary transverse, longitudinal, and angular velocities. Under these control strategies, the resulting closed-loop coupled system has been proven to be well-posed using the nonlinear semigroup approach. Furthermore, the study employs a combination of a generalized Gronwall-type integral inequality and the integral multiplier method to investigate the explicit decay rate of the closed-loop coupled system. This investigation involves the construction of a novel energy functional. Numerical simulations validate the effectiveness of the proposed controls, with the explicit exponential decay rate closely mirroring the actual decay rate.