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Control and Stabilization of Degenerate Wave Equations

2017; Society for Industrial and Applied Mathematics; Volume: 55; Issue: 3 Linguagem: Inglês

10.1137/15m1020538

1095-7138

Fatiha Alabau‐Boussouira, Piermarco Cannarsa, Günter Leugering,

Numerical methods in inverse problems

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 prove the blowup of the observability time when $\mu_a$ converges to $2$ from below. Thus, using the Hilbert uniqueness 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 [F. Alabau-Boussouira, Appl. Math. Optim., 51 (2005), pp. 61--105], [F. Alabau-Boussouira, J. Differential Equations, 249 (2010), pp. 1473--1517], together with our results for linear damping, to perform this stability analysis.