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In this paper, we study the boundary penalty method for optimal control of unsteady Navier-Stokes type system that has been proposed as an alternative for Dirichlet boundary control. Existence and uniqueness of solutions are demonstrated and existence of optimal control for a class of optimal control problems is established. The asymptotic behavior of solution, with respect to the penalty parameter ϵ, is studied. In particular, we prove convergence of solutions of penalized control problem to the corresponding solutions of the Dirichlet control problem, as the penalty parameter goes to zero. We also derive an optimality system and determine optimal solutions. In order to illustrate the theoretical results and the practical utility of control, we numerically address the problem of controlling unsteady convection with Soret effect using a gradient-based method. Numerical results show the effectiveness of the approach.
@article{COCV_2014__20_3_840_0,
author = {Ravindran, S. S.},
title = {Dirichlet control of unsteady {Navier-Stokes} type system related to {Soret} convection by boundary penalty method},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {840--863},
publisher = {EDP-Sciences},
volume = {20},
number = {3},
year = {2014},
doi = {10.1051/cocv/2013086},
mrnumber = {3264226},
zbl = {1301.35093},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013086/}
}
TY - JOUR AU - Ravindran, S. S. TI - Dirichlet control of unsteady Navier-Stokes type system related to Soret convection by boundary penalty method JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 840 EP - 863 VL - 20 IS - 3 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013086/ DO - 10.1051/cocv/2013086 LA - en ID - COCV_2014__20_3_840_0 ER -
%0 Journal Article %A Ravindran, S. S. %T Dirichlet control of unsteady Navier-Stokes type system related to Soret convection by boundary penalty method %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 840-863 %V 20 %N 3 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013086/ %R 10.1051/cocv/2013086 %G en %F COCV_2014__20_3_840_0
Ravindran, S. S. Dirichlet control of unsteady Navier-Stokes type system related to Soret convection by boundary penalty method. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 3, pp. 840-863. doi: 10.1051/cocv/2013086
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