Convergence of distributed optimal controls in mixed elliptic problem by the penalization method

C.M. GARIBOLDI - D.A. TARZIA

MATHEMATICAE NOTAE

Univ. Nac. de Rosario

Lugar: Rosario; Año: 2008 vol. 45 p. 1 - 19

0025-553X

We consider a distributed optimal control problem $(P)$ and asequence of distributed optimal control problems $(P_{alpha })$,for each $alpha >0$ (heat transfer coefficient). Both problems arerelated with steady-state heat conduction problems for the samePoisson equation with different mixed boundary condition for $(P)$and $(P_{alpha })$ respectively. We use the penalization method inorder to obtain a family of optimal control problems $(P_{epsilon})$ for each $epsilon >0$ and a family $(P_{alpha epsilon })$ forfixed $alpha >0$ with their correspondent cost functions$J_{epsilon }$ and $J_{alpha epsilon }$. We prove strongconvergence as $epsilon ightarrow 0$ of the optimal control$g_{epsilon }$ of $(P_{epsilon })$ to the optimal control $g$ of$(P)$ and of the system state $U_{epsilon }$ of problem$(P_{epsilon })$ to the system state $U$ of problem $(P)$ insuitable Sobolev spaces. We obtain similar results for fixed $alpha>0$ and $epsilon ightarrow 0$ in relation for the problems$(P_{alpha })$ and $(P_{alpha epsilon }).$ Finally, we obtainweak convergence of solutions of the problems $(P_{alpha epsilon})$ to the solution of the problem $(P_{epsilon })$ when $alphaightarrow infty $, for fixed $epsilon >0$. This result can beconsidered as a new proof of the convergence obtained inGariboldi-Tarzia, Appl. Math. Optim., 47 (2003), 213-230 by usingthe variational equality theory.