Stabilization of the wave equation through nonlinear Dirichlet actuation

¹ Mathematics and Statistics, Faculty of Information Technology and Communication Sciences, Tampere University, PO Box 692, 33101 Tampere, Finland
² Department of Engineering, University of Perugia, 06125 Perugia, Italy
³ Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, 38000 Grenoble, France

^* Corresponding author: nicolas.vanspranghe@tuni.fi

Received: 4 October 2021
Accepted: 18 November 2022

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 L²(Ω) × H⁻¹(Ω). 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 H¹(Ω) × L²(Ω) of the wave equation with Neumann boundary conditions.