Exponential Stabilization of the Uncertain Wave Equation via Distributed Dynamic Input Extension

The wave equation is studied under the effect of a persistent external perturbation. A dynamical distributed controller is suggested, based on an infinite-dimensional generalization of second-order sliding-mode control techniques, that provides the exponential attainment of a sufficiently smooth arbitrary reference of the state trajectory. The control system comprises of both feedforward and feedback parts, the latter being a discontinuous term directly connected to the plant input through a dynamical filter that augments the system state smoothing out the discontinuity of the feedback control loop. As a result, a continuous input is applied to the plant. A constructive Lyapunov-based proof of convergence of the proposed control algorithm is carried out and supporting numerical results are presented.