Stabilization of the wave equation through nonlinear Dirichlet actuation

TL;DR

This paper studies the nonlinear stabilization of the wave equation with Dirichlet boundary conditions, proving convergence to zero and establishing polynomial decay rates under certain conditions, extending known results to Dirichlet cases.

Contribution

It provides the first Dirichlet boundary condition results for nonlinear wave stabilization with polynomial decay rates, complementing existing Neumann boundary results.

Findings

Solutions converge to zero in energy space.

Polynomial decay rates are established for smooth initial data.

Results extend nonlinear stabilization theory to Dirichlet boundary conditions.

Abstract

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 $L^2(\Omega) \times H^{-1}(\Omega)$. 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 $H^1(\Omega)\times L^2(\Omega)$ of the wave equation with Neumann boundary conditions.