Geodesic
Singular optimal control of a 1-D parabolic-hyperbolic degenerate equation
Gueye, Mamadou1 ; Lissy, Pierre2 
1 Departamento de Matematica, Universidad Tecnica Federico Santa Maria, Casilla 110-V, Valparaiso, Chile.
2 CEREMADE, UMR 7534, Université Paris-Dauphine & CNRS, Place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France.
ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 4, pp. 1184-1203

Voir la notice de l'article provenant de la source Numdam

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.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2016036
Classification : 35K65, 93B05, 30E05
Keywords: Degenerate parabolic equation, cost of the control, uniform controllability, Bessel functions

Gueye, Mamadou 1 ; Lissy, Pierre 2

1 Departamento de Matematica, Universidad Tecnica Federico Santa Maria, Casilla 110-V, Valparaiso, Chile.
2 CEREMADE, UMR 7534, Université Paris-Dauphine & CNRS, Place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France. 
@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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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, Tome 22 (2016) no. 4, pp. 1184-1203. doi: 10.1051/cocv/2016036

Cité par Sources :