Stabilization of One-Dimensional Wave Equation With Nonlinear Boundary Condition Subject to Boundary Control Matched Disturbance

Abstract

In this paper, we consider the stabilization of a one-dimensional wave equation with nonlinear van der Pol type boundary condition that covers the antistable boundary, and subject to boundary control matched disturbance on the other side. Due to the nonlinear boundary condition and disturbance, the uncontrolled system may present spatiotemporal chaotic, period-doubling bifurcation, and some other dynamical behaviors. We will deal with this disturbance, which is supposed to be bounded only, by the integral sliding mode control. The well-posedness of the system for the closed-loop system is proved and the “reaching condition” is obtained. Finally, we provide some numerical simulations to illustrate the theoretical outcomes.