Stabilization of the wave equation through nonlinear Dirichlet actuation

In this paper, we consider the problem of nonlinear (in particular, saturated) stabilization of the high-dimensional wave equation with Dirichlet boundary conditions. The wave dynamics are subject to a dissipative nonlinear velocity feedback and generate a strongly continuous semigroup of contractions on the optimal energy space $L^2(\mathrm{\Omega })\times H^{-1}(\mathrm{\Omega })$. It is first proved that any solution to the closed-loop equations converges to zero in the aforementioned topology. Secondly, under the condition that the feedback nonlinearity has linear growth around zero, polynomial energy decay rates are established for solutions with smooth initial data. This constitutes new Dirichlet counterparts to well-known results pertaining to nonlinear stabilization in $H^1(\mathrm{\Omega })\times L^2(\mathrm{\Omega })$ of the wave equation with Neumann boundary conditions.