Dirichlet control of unsteady Navier–Stokes type system related to Soret convection by boundary penalty method

S. S. Ravindran

ESAIM: Control, Optimisation and Calculus of Variations (2014)

  • Volume: 20, Issue: 3, page 840-863
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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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 20.3 (2014): 840-863. <http://eudml.org/doc/272927>.

@article{Ravindran2014,
abstract = {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.},
author = {Ravindran, S. S.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {boundary penalty; dirichlet boundary control; Navier–stokes type system; soret convection; Navier-Stokes type system; Soret convection; Dirichlet boundary control},
language = {eng},
number = {3},
pages = {840-863},
publisher = {EDP-Sciences},
title = {Dirichlet control of unsteady Navier–Stokes type system related to Soret convection by boundary penalty method},
url = {http://eudml.org/doc/272927},
volume = {20},
year = {2014},
}

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
PB - EDP-Sciences
VL - 20
IS - 3
SP - 840
EP - 863
AB - 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.
LA - eng
KW - boundary penalty; dirichlet boundary control; Navier–stokes type system; soret convection; Navier-Stokes type system; Soret convection; Dirichlet boundary control
UR - http://eudml.org/doc/272927
ER -

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