High‐gain stabilization for an axially moving beam under boundary feedback control

This paper establishes the global existence and high‐gain stabilization of a nonlinear axially moving beam with control input at the free boundary. A high‐gain controller based on the transverse velocity feedbacks of the moving beam at the free end is designed. The existence and uniqueness of the solution depending on the initial values continuously for the resulting closed‐loop system are established by invoking the Faedo–Galerkin approximation approach. Then constructing a novel energy‐like function, the explicit exponential decay rate of the closed‐loop system is obtained via a generalized Gronwall‐type integral inequality.