Convergence of boundary optimal controls in mixed elliptic problems

C.M. GARIBOLDI – D.A. TARZIA

Proceedings on Applied Mathematics and Mechanics

Wiley Interscience

Lugar: New York; Año: 2007 vol. 7 p. 1060403 - 1060404

1617-7061

We consider a steady-state heat conduction problem ${P}_{alpha ,}$with mixed boundary conditions for the Poisson equation in a bounded multidimensional domain $Omega $ depending of a positive parameter $alpha $ which represents the heat transfer coefficient on a portion hinspace $Gamma _{1}$ of the boundary of $Omega $. We consider, for each $alpha >0,$ a cost function $J_{alpha }$ and we formulate boundary optimal control problems with restrictions over the heat flux $q,$on a complementary portion $Gamma _{2}$ of the boundary of $Omega $ . We obtain that the optimality conditions are given by a complementary free boundary problem in $Gamma _{2}$ in terms of the adjoint state. We prove that the optimal control $q_{op_{alpha }}$ and its corresponding system state $u_{q_{op_{alpha }}alpha }$ and adjoint state $ p_{q_{op_{alpha }}alpha }$ for each $alpha$ are strongly convergent to $q_{op},$ $u_{q_{op}}$ and ${p}_{q_{op}}$ in $L^{2}(Gamma _{2})$, $H^{1}(Omega )$, and $H^{1}(Omega )$ respectively when $alpha ightarrow infty $. We also prove that these limit functions are respectively the optimal control, the system state and the adjoint state corresponding to another boundary optimal control problem with restrictions for the same Poisson equation with a different boundary condition$,$on the portion $Gamma _{1},$. We use the elliptic variational inequality theory in order to prove all the strong convergences. In this paper, we generalize the convergence result obtained in Ben Belgacem-El Fekih-Metoui, ESAIM:M2AN, 37 (2003), 833-850 by considering boundary optimal control problems with restrictions on the heat flux $q$ defined on $Gamma _{2}$ and the parameter $alpha $ (which goes to $inf $inity) is defined on $Gamma _{1}$.