# Control problems for convection-diffusion equations with control localized on manifolds

• Volume: 6, page 467-488
• ISSN: 1292-8119

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## Abstract

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We consider optimal control problems for convection-diffusion equations with a pointwise control or a control localized on a smooth manifold. We prove optimality conditions for the control variable and for the position of the control. We do not suppose that the coefficient of the convection term is regular or bounded, we only suppose that it has the regularity of strong solutions of the Navier–Stokes equations. We consider functionals with an observation on the gradient of the state. To obtain optimality conditions we have to prove that the trace of the adjoint state on the control manifold belongs to the dual of the control space. To study the state equation, which is an equation with measures as data, and the adjoint equation, which involves the divergence of ${L}^{p}$-vector fields, we first study equations without convection term, and we next use a fixed point method to deal with the complete equations.

## How to cite

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Nguyen, Phuong Anh, and Raymond, Jean-Pierre. "Control problems for convection-diffusion equations with control localized on manifolds." ESAIM: Control, Optimisation and Calculus of Variations 6 (2001): 467-488. <http://eudml.org/doc/90603>.

@article{Nguyen2001,
abstract = {We consider optimal control problems for convection-diffusion equations with a pointwise control or a control localized on a smooth manifold. We prove optimality conditions for the control variable and for the position of the control. We do not suppose that the coefficient of the convection term is regular or bounded, we only suppose that it has the regularity of strong solutions of the Navier–Stokes equations. We consider functionals with an observation on the gradient of the state. To obtain optimality conditions we have to prove that the trace of the adjoint state on the control manifold belongs to the dual of the control space. To study the state equation, which is an equation with measures as data, and the adjoint equation, which involves the divergence of $L^p$-vector fields, we first study equations without convection term, and we next use a fixed point method to deal with the complete equations.},
author = {Nguyen, Phuong Anh, Raymond, Jean-Pierre},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {pointwise control; optimal control; convection-diffusion equation; control localized on manifolds; regularity; Navier-Stokes equations; optimality conditions},
language = {eng},
pages = {467-488},
publisher = {EDP-Sciences},
title = {Control problems for convection-diffusion equations with control localized on manifolds},
url = {http://eudml.org/doc/90603},
volume = {6},
year = {2001},
}

TY - JOUR
AU - Nguyen, Phuong Anh
AU - Raymond, Jean-Pierre
TI - Control problems for convection-diffusion equations with control localized on manifolds
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2001
PB - EDP-Sciences
VL - 6
SP - 467
EP - 488
AB - We consider optimal control problems for convection-diffusion equations with a pointwise control or a control localized on a smooth manifold. We prove optimality conditions for the control variable and for the position of the control. We do not suppose that the coefficient of the convection term is regular or bounded, we only suppose that it has the regularity of strong solutions of the Navier–Stokes equations. We consider functionals with an observation on the gradient of the state. To obtain optimality conditions we have to prove that the trace of the adjoint state on the control manifold belongs to the dual of the control space. To study the state equation, which is an equation with measures as data, and the adjoint equation, which involves the divergence of $L^p$-vector fields, we first study equations without convection term, and we next use a fixed point method to deal with the complete equations.
LA - eng
KW - pointwise control; optimal control; convection-diffusion equation; control localized on manifolds; regularity; Navier-Stokes equations; optimality conditions
UR - http://eudml.org/doc/90603
ER -

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