We study the strong stabilization of wave equations on some sphere-like manifolds, with rough damping terms which do not satisfy the geometric control condition posed by Rauch−Taylor [J. Rauch and M. Taylor, Commun. Pure Appl. Math. 28 (1975) 501–523] and Bardos−Lebeau−Rauch [C. Bardos, G. Lebeau and J. Rauch, SIAM J. Control Optimiz. 30 (1992) 1024–1065]. We begin with an unpublished result of G. Lebeau, which states that on ${\mathbb{S}}^{d}$, the indicator function of the upper hemisphere strongly stabilizes the damped wave equation, even though the equators, which are geodesics contained in the boundary of the upper hemisphere, do not enter the damping region. Then we extend this result on dimension $2$, to Zoll surfaces of revolution, whose geometry is similar to that of ${\mathbb{S}}^{2}$. In particular, geometric objects such as the equator, and the hemi-surfaces are well defined. Our result states that the indicator function of the upper hemi-surface strongly stabilizes the damped wave equation, even though the equator, as a geodesic, does not enter the upper hemi-surface either.

Keywords: Wave equation, semiclassical analysis, control theory, geodesic flow

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@article{COCV_2018__24_4_1759_0, author = {Zhu, Hui}, title = {Stabilization of damped waves on spheres and {Zoll} surfaces of revolution}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1759--1788}, publisher = {EDP-Sciences}, volume = {24}, number = {4}, year = {2018}, doi = {10.1051/cocv/2017073}, zbl = {1421.35032}, mrnumber = {3922432}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2017073/} }

TY - JOUR AU - Zhu, Hui TI - Stabilization of damped waves on spheres and Zoll surfaces of revolution JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1759 EP - 1788 VL - 24 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017073/ DO - 10.1051/cocv/2017073 LA - en ID - COCV_2018__24_4_1759_0 ER -

%0 Journal Article %A Zhu, Hui %T Stabilization of damped waves on spheres and Zoll surfaces of revolution %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1759-1788 %V 24 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017073/ %R 10.1051/cocv/2017073 %G en %F COCV_2018__24_4_1759_0

Zhu, Hui. Stabilization of damped waves on spheres and Zoll surfaces of revolution. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 4, pp. 1759-1788. doi : 10.1051/cocv/2017073. http://archive.numdam.org/articles/10.1051/cocv/2017073/

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