Convergence of boundary optimal controls in mixed elliptic problems

Abstract

AbstractWe consider a steady‐state heat conduction problem Pα withmixed boundary conditions for the Poisson equation in a bounded multidimensional domain Ω depending of a positive parameter α which represents the heat transfer coefficient on a portion Γ1 of the boundary of Ω. We consider, for each α > 0, a cost function Jα and we formulate boundary optimal control problems with restrictions over the heat flux q on a complementary portion Γ2 of the boundary of Ω. We obtain that the optimality conditions are given by a complementary free boundary problem in Γ2 in terms of the adjoint state. We prove that the optimal control q and its corresponding system state u and adjoint state p for each α are strongly convergent to qop, u and p in L2(Γ2), H1(Ω), and H1(Ω) respectively when α → ∞. 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 Γ1. 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 Γ2 and the parameter α (which goes to infinity) is defined on Γ1. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)