Control and stabilization of degenerate wave equations

TL;DR

This paper investigates the controllability and stabilization of a one-dimensional wave equation with a boundary-degenerate diffusion coefficient, establishing observability inequalities, controllability results, and exponential stabilization methods for various degeneracy levels.

Contribution

It provides new observability inequalities and controllability results for degenerate wave equations, including both weak and strong degeneracy cases, and extends stabilization techniques to nonlinear degenerate wave systems.

Findings

Observability inequalities established for weakly and strongly degenerate equations.

Exact controllability achieved for degeneracy parameter b1a0b1a0<2.

Exponential boundary stabilization proven for degenerate wave equations.

Abstract

We study a wave equation in one space dimension with a general diffusion coefficient which degenerates on part of the boundary. Degeneracy is measured by a real parameter $\mu_a>0$. We establish observability inequalities for weakly (when $\mu_a \in [0,1[$) as well as strongly (when $\mu_a \in [1,2[$) degenerate equations. We also prove a negative result when the diffusion coefficient degenerates too violently (i.e. when $\mu_a>2$) and the blow-up of the observability time when $\mu_a$ converges to $2$ from below. Thus, using the HUM method we deduce the exact controllability of the corresponding degenerate control problem when $\mu_a \in [0,2[$. We conclude the paper by studying the boundary stabilization of the degenerate linearly damped wave equation and show that a suitable boundary feedback stabilizes the system exponentially. We extend this stability analysis to the degenerate nonlinearly boundary damped wave equation, for an arbitrarily growing nonlinear feedback close to the origin. This analysis proves that the degeneracy does not affect the optimal energy decay rates at large time. We apply the optimal-weight convexity method of \cite{alaamo2005, alajde2010} together with the results of the previous section, to perform this stability analysis.