Robustness to In-Domain Viscous Damping of a Collocated Boundary Adaptive Feedback Law for an Antidamped Boundary Wave PDE

In this paper, the robustness to model mismatch of a preexisting collocated boundary adaptive feedback law is investigated. This control law was originally designed for an antidamped pure wave partial differential equation (PDE). Actuation and measurements are located at the same boundary. Adaptive terms account for uncertain parameters located at the antidamped boundary, opposite to the collocated actuation and measurement. By extending and transforming the system state using, in particular, backstepping, this paper establishes that this controller is robust to sufficiently small in-domain damping. In particular, both stability and attractivity (convergence) are established similarly as in the nominal case. Note moreover that, assuming that some parameters are known, the exponential stability to an attractor holds. Simulations are performed to illustrate the interest of this study to attenuate mechanical vibrations in an oil-drilling context.