PI output feedback for the wave PDE with second order dynamical boundary conditions

The paper focuses on a one-dimensional wave equation subject to second order dynamic boundary conditions with source terms on a Hilbert space. The purpose is the regulation of the wave velocity toward a given constant by the means of a Proportional-Integral action. The regulation problem is reformulated as a stability problem over an attractor which is not bounded in the state space. The asymptotic stability of which is established using La Salle invariance principle in Hilbert space. In order to proceed, this paper proves that the evolution problem associated with a maximal linear operator (or equivalently linear
$C_{0}$
-semigroup of contraction) gives pre compact trajectories. This result is used on the considered wave equation on the state space-attractor quotient. The state space-attractor quotient is proven to be a Hilbert space.