Numerical solution of a parabolic optimal control problem with point-wise state constraints
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 158 (2016) no. 1, pp. 81-89
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The problem of optimal control over the system governed by the Dirichlet boundary value problem for a linear parabolic equation is constructed. Point-wise constraints are imposed on both control and state functions. The right-hand side of the equation is a control function in the problem. The objective functional contains an observation which is distributed in the space-time domain. Finite-difference approximation is constructed for the optimal control problem based on the Euler forward scheme for the state parabolic equation. The existence of its unique solution is proved. Constrained saddle point problem corresponding to the mesh optimal control problem is constructed. The existence of a solution for this saddle point problem and the convergence of the generalized Uzawa iterative method are proved. The results of numerical experiments are given.
Keywords:
optimal control, parabolic state equation, state constraints, finite-difference approximation, iterative method.
@article{UZKU_2016_158_1_a5,
author = {A. V. Lapin and A. A. Platonov},
title = {Numerical solution of a~parabolic optimal control problem with point-wise state constraints},
journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
pages = {81--89},
publisher = {mathdoc},
volume = {158},
number = {1},
year = {2016},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/UZKU_2016_158_1_a5/}
}
TY - JOUR AU - A. V. Lapin AU - A. A. Platonov TI - Numerical solution of a parabolic optimal control problem with point-wise state constraints JO - Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki PY - 2016 SP - 81 EP - 89 VL - 158 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/UZKU_2016_158_1_a5/ LA - ru ID - UZKU_2016_158_1_a5 ER -
%0 Journal Article %A A. V. Lapin %A A. A. Platonov %T Numerical solution of a parabolic optimal control problem with point-wise state constraints %J Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki %D 2016 %P 81-89 %V 158 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/item/UZKU_2016_158_1_a5/ %G ru %F UZKU_2016_158_1_a5
A. V. Lapin; A. A. Platonov. Numerical solution of a parabolic optimal control problem with point-wise state constraints. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 158 (2016) no. 1, pp. 81-89. http://geodesic.mathdoc.fr/item/UZKU_2016_158_1_a5/