Robustness of an adaptive output feedback for an anti-damped boundary wave PDE in presence of in-domain viscous damping

This paper presents a robustness result for the use of a previously developed adaptive output-feedback controller designed for a pure wave PDE. This one-dimensional wave PDE has a boundary actuation opposite to an unknown anti-damping dynamics boundary. We prove that stabilization is still valid in the case of in-domain viscous damping using a controller designed neglecting the damping, provided that the damping coefficient is small enough. Our robustness proof grounds on the use of an alternative pure wave PDE, neglecting the indomain damping. We compare these two systems and apply a tailored backstepping transformation to carry out Lyapunov analysis. Numerical simulations show the relevance of our result for drilling applications and illustrate the merits of this control design.