In this paper, we consider the controllability of a strongly degenerate parabolic equation with a degenerate one-order transport term. Despite the strong degeneracy, we prove a result of well-posedness and null controllability with a Dirichlet boundary control that acts on the degenerate part of the boundary. Then, we study the uniform controllability in the vanishing viscosity limit and prove that the cost of the control explodes exponentially fast in small time and converges exponentially fast in large time in some adapted weighted norm. The main tools used are a spectral decomposition involving Bessel functions and their zeros, some usual results on admissibility of scalar controls for diagonal semigroups, and the moment method of Fattorini and Russell.
Accepted:
DOI: 10.1051/cocv/2016036
Mots-clés : Degenerate parabolic equation, cost of the control, uniform controllability, Bessel functions
@article{COCV_2016__22_4_1184_0, author = {Gueye, Mamadou and Lissy, Pierre}, title = {Singular optimal control of a {1-D} parabolic-hyperbolic degenerate equation}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1184--1203}, publisher = {EDP-Sciences}, volume = {22}, number = {4}, year = {2016}, doi = {10.1051/cocv/2016036}, zbl = {1357.35202}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2016036/} }
TY - JOUR AU - Gueye, Mamadou AU - Lissy, Pierre TI - Singular optimal control of a 1-D parabolic-hyperbolic degenerate equation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 1184 EP - 1203 VL - 22 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016036/ DO - 10.1051/cocv/2016036 LA - en ID - COCV_2016__22_4_1184_0 ER -
%0 Journal Article %A Gueye, Mamadou %A Lissy, Pierre %T Singular optimal control of a 1-D parabolic-hyperbolic degenerate equation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 1184-1203 %V 22 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016036/ %R 10.1051/cocv/2016036 %G en %F COCV_2016__22_4_1184_0
Gueye, Mamadou; Lissy, Pierre. Singular optimal control of a 1-D parabolic-hyperbolic degenerate equation. ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 4, pp. 1184-1203. doi : 10.1051/cocv/2016036. http://archive.numdam.org/articles/10.1051/cocv/2016036/
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