Boundary control of wave equations with input saturation and boundary displacement constraints

This paper investigates the boundary control problem of a wave equation system with dual constraints: input saturation and boundary displacement constraints at the L-position. To address the challenges posed by nonlinear input saturation and boundary displacement limitations, a shifting function and Nussbaum gain method are introduced to design the boundary control law. By constructing an energy function that incorporates a cross term and a barrier Lyapunov function, the closed-loop system stability is proven. The proposed control strategy ensures that the boundary displacement enters a predefined constraint range within a preset finite time while guaranteeing the input remains within saturation limits. Numerical simulations are conducted to validate the effectiveness of the proposed method. The results demonstrate that the system displacement and control input satisfy the design constraints, and the system achieves an asymptotic stabilization.