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

Phuong Anh Nguyen; Jean-Pierre Raymond

ESAIM: Control, Optimisation and Calculus of Variations (2010)

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

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topNguyen, 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 (2010): 467-488. <http://eudml.org/doc/197296>.

@article{Nguyen2010,

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 Lp-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.; pointwise control; control localized on manifolds; regularity; Navier-Stokes equations; optimality conditions},

language = {eng},

month = {3},

pages = {467-488},

publisher = {EDP Sciences},

title = {Control problems for convection-diffusion equations with control localized on manifolds},

url = {http://eudml.org/doc/197296},

volume = {6},

year = {2010},

}

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

DA - 2010/3//

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 Lp-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.; pointwise control; control localized on manifolds; regularity; Navier-Stokes equations; optimality conditions

UR - http://eudml.org/doc/197296

ER -

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