In this paper, we consider the optimal control of semilinear elliptic PDEs with random inputs. These problems are often nonconvex, infinite-dimensional stochastic optimization problems for which we employ risk measures to quantify the implicit uncertainty in the objective function. In contrast to previous works in uncertainty quantification and stochastic optimization, we provide a rigorous mathematical analysis demonstrating higher solution regularity (in stochastic state space), continuity and differentiability of the control-to-state map, and existence, regularity and continuity properties of the control-to-adjoint map. Our proofs make use of existing techniques from PDE-constrained optimization as well as concepts from the theory of measurable multifunctions. We illustrate our theoretical results with two numerical examples motivated by the optimal doping of semiconductor devices.
Mots-clés : Risk-averse, PDE-constrained optimization, semilinear PDEs, uncertainty quantification, stochastic optimization, measurable multifunctions
@article{COCV_2020__26_1_A53_0,
author = {Kouri, D.P. and Surowiec, T.M.},
title = {Risk-averse optimal control of semilinear elliptic {PDEs}},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {26},
year = {2020},
doi = {10.1051/cocv/2019061},
mrnumber = {4145243},
zbl = {1448.49006},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2019061/}
}
TY - JOUR AU - Kouri, D.P. AU - Surowiec, T.M. TI - Risk-averse optimal control of semilinear elliptic PDEs JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2019061/ DO - 10.1051/cocv/2019061 LA - en ID - COCV_2020__26_1_A53_0 ER -
%0 Journal Article %A Kouri, D.P. %A Surowiec, T.M. %T Risk-averse optimal control of semilinear elliptic PDEs %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2019061/ %R 10.1051/cocv/2019061 %G en %F COCV_2020__26_1_A53_0
Kouri, D.P.; Surowiec, T.M. Risk-averse optimal control of semilinear elliptic PDEs. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 53. doi : 10.1051/cocv/2019061. http://www.numdam.org/articles/10.1051/cocv/2019061/
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Cité par Sources :
DPK’s research was sponsored by DARPA EQUiPS grant SNL 014150709.
TMS’s research was sponsored by DFG grant no. SU 963/1-1 “Generalized Nash Equilibrium Problems with Partial Differential Operators: Theory, Algorithms, and Risk Aversion”.
Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.
