Appl Math Optim 47:213–230 (2003)
2003 Springer-Verlag New York Inc.
Convergence of Distributed Optimal Controls on the
Internal Energy in Mixed Elliptic Problems when the
Heat Transfer Coefﬁcient Goes to Inﬁnity
Claudia M. Gariboldi
and Domingo A. Tarzia
Departamento Matem´atica, FCEFQyN, Univ. Nac. de R´ıo Cuarto,
Ruta 36 Km 601, 5800 R´ıo Cuarto, Argentina
Departamento Matem´atica-CONICET, FCE, Univ. Austral,
Paraguay 1950, S2000FZF Rosario, Argentina
Communicated by A. Bensoussan
Abstract. We consider a steady-state heat conduction problem P
boundary conditions for the Poisson equation depending on a positive parameter α,
which represents the heat transfer coefﬁcient on a portion
of the boundary of a
given bounded domain in R
. We formulate distributed optimal control problems
over the internal energy g for each α. We prove that the optimal control g
its corresponding system u
and adjoint p
states for each α are strongly
convergent to g
, respectively, in adequate functional spaces. We also
prove that these limit functions are respectively the optimal control, and the system
and adjoint states corresponding to another distributed optimal control problem for
the same Poisson equation with a different boundary condition on the portion
We use the ﬁxed point and elliptic variational inequality theories.
Key Words. Variational inequality, Distributed optimal control, Mixed elliptic
problem, Adjoint state, Steady-state Stefan problem, Optimality condition, Fixed
AMS Classiﬁcation. 49J20, 35J85, 35R35.
This paper has been partially sponsored by the Project “Free Boundary Problems for the Heat-Diffusion
Equation” from CONICET - UA, Rosario (Argentina).