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

Université Paul Sabatier, UMR CNRS MIP, UFR MIG, 31062 Toulouse Cedex 4, France; nguyen@mip.ups-tlse.fr. and

Received: 31 May 2000
Revised: 15 February 2001

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.