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

Published online by Cambridge University Press:  15 August 2002

Phuong Anh Nguyen  and
Jean-Pierre Raymond
Phuong Anh Nguyen
Affiliation:
Université Paul Sabatier, UMR CNRS MIP, UFR MIG, 31062 Toulouse Cedex 4, France; nguyen@mip.ups-tlse.fr. and
Jean-Pierre Raymond
Affiliation:
Université Paul Sabatier, UMR CNRS MIP, UFR MIG, 31062 Toulouse Cedex 4, France; nguyen@mip.ups-tlse.fr. and
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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.

Keywords

Pointwise controloptimal controlconvection-diffusion equationcontrol localized on manifolds.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 6 , 2001 , pp. 467 - 488
Copyright
© EDP Sciences, SMAI, 2001

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