Optimal control problems without control costs in general do not possess solutions due to the lack of coercivity. However, unilateral constraints together with the assumption of existence of strictly positive solutions of a pre-adjoint state equation, are sufficient to obtain existence of optimal solutions in the space of Radon measures. Optimality conditions for these generalized minimizers can be obtained using Fenchel duality, which requires a non-standard perturbation approach if the control-to-observation mapping is not continuous (e.g., for Neumann boundary control in three dimensions). Combining a conforming discretization of the measure space with a semismooth Newton method allows the numerical solution of the optimal control problem.
Accepté le :
DOI : 10.1051/cocv/2015046
Mots-clés : Optimal control, measure control, control constraints, Fenchel duality, unbounded operators
@article{COCV_2017__23_1_217_0,
author = {Clason, Christian and Schiela, Anton},
title = {Optimal control of elliptic equations with positive measures},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {217--240},
publisher = {EDP-Sciences},
volume = {23},
number = {1},
year = {2017},
doi = {10.1051/cocv/2015046},
mrnumber = {3601022},
zbl = {1365.49005},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2015046/}
}
TY - JOUR AU - Clason, Christian AU - Schiela, Anton TI - Optimal control of elliptic equations with positive measures JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 217 EP - 240 VL - 23 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2015046/ DO - 10.1051/cocv/2015046 LA - en ID - COCV_2017__23_1_217_0 ER -
%0 Journal Article %A Clason, Christian %A Schiela, Anton %T Optimal control of elliptic equations with positive measures %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 217-240 %V 23 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2015046/ %R 10.1051/cocv/2015046 %G en %F COCV_2017__23_1_217_0
Clason, Christian; Schiela, Anton. Optimal control of elliptic equations with positive measures. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 1, pp. 217-240. doi : 10.1051/cocv/2015046. http://www.numdam.org/articles/10.1051/cocv/2015046/
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