Boundary Stabilization of the linear MGT equation with Feedback Neumann control

TL;DR

This paper demonstrates boundary feedback stabilization of the linearized Jordan-Moore-Gibson-Thompson equation in the critical case, achieving uniform exponential stability even with degenerate viscoelasticity parameters, advancing control theory for nonlinear acoustic models.

Contribution

It introduces a boundary feedback stabilization method for the linear MGT equation in the critical case, with stability results uniform across degenerate viscoelasticity parameters.

Findings

Achieved uniform exponential stability with boundary feedback.

Stability holds even when viscoelasticity parameter is zero.

Provides a foundation for optimal boundary control on infinite horizon.

Abstract

The Jordan-Moore-Gibson-Thompson (JMGT)\cite{christov_heat_2005,jordan_nonlinear_2008,straughan_heat_2014} equation is a benchmark model describing propagation of nonlinear acoustic waves in heterogeneous fluids at rest. This is a third-order (in time) dynamics which accounts for a finite speed of propagation of heat signals (see \cite{coulouvrat_equations_1992,crighton_model_1979,jordan_nonlinear_2008,jordan_second-sound_2014,kaltenbacher_jordan-moore-gibson-thompson_2019}). In this paper, we study a boundary stabilization of linearized version (also known as MGT-equation) in the {\it critical case}, configuration in which the smallness of the diffusion effects leads to conservative dynamics \cite{kaltenbacher_wellposedness_2011}. Through a single measurement in {\it{feedback}} form made on a non-empty, relatively open portion of the boundary under natural geometric conditions, we were able to obtain uniform exponential stability results that are, in addition, uniform with respect to the space-dependent viscoelasticity parameter which no longer needs to be assumed positive and in fact can be degenerate and taken to be zero on the whole domain. This result, of independent interest in the area of boundary stabilization of MGT equations, provides a necessary first step for the study of optimal boundary feedback control on {\it infinite horizon} \cite{bucci_feedback_2019}.