Adaptive Boundary Control of an Euler–Bernoulli Overhead Bridge under Moving Vehicle Loads with Uncertain Boundary Stiffness: Explicit Incorporation of Disturbance Effects in the Lyapunov Stability Analysis

This study examines the adaptive boundary control of an Euler–Bernoulli overhead bridge subjected to moving vehicle loads under uncertain boundary stiffness, with particular attention to the influence of external disturbances on Lyapunov stability. A Galerkin reduced-order model based on the fundamental mode is used to describe the bridge dynamics, while an adaptive boundary-control strategy is developed to compensate for boundary-stiffness uncertainty and reduce vibration caused by the moving vehicle. The vehicle load is incorporated explicitly as a time-dependent bounded disturbance, and a disturbance-inclusive Lyapunov approach is employed to establish the uniform ultimate boundedness (UUB) of the closed-loop system. Numerical results show that the vehicle reaches the 10 m bridge exit at approximately, producing the most significant transient response, after which the modal displacement, velocity, control effort, and Lyapunov measures decrease rapidly as the disturbance leaves the bridge. The corresponding spatial deflection profiles confirm the effective attenuation of the bridge response following vehicle departure, while the Lyapunov analysis indicates that the controlled states remain bounded and gradually approach the desired operating condition. Overall, the findings demonstrate that the proposed adaptive boundary-control strategy can effectively reduce transient vibrations and maintain stable bridge dynamics despite uncertain boundary stiffness, thereby providing numerical support for the disturbance-inclusive UUB stability framework.