  We consider a steady-state heat conduction problem in a multidimensional
bounded domain Omega for the Poisson equation with constant internal energy g
and mixed boundary conditions given by a constant temperature b in the portion
Gamma_1 of the boundary and a constant heat flux q in the remaining portion
Gamma_2 of the boundary. Moreover, we consider a family of steady-state heat
conduction problems with a convective condition on the boundary Gamma_1 with
heat transfer coefficient alpha and external temperature b. We obtain
explicitly, for a rectangular domain in R^2, an annulus in R^2 and a spherical
shell in R^3, the optimal controls, the system states and adjoint states for
the following optimal control problems: a distributed control problem on the
internal energy g, a boundary optimal control problem on the heat flux q, a
boundary optimal control problem on the external temperature b and a
distributed-boundary simultaneous optimal control problem on the source g and
the flux q. These explicit solutions can be used for testing new numerical
methods as a benchmark test. In agreement with theory, it is proved that the
system state, adjoint state, optimal controls and optimal values corresponding
to the problem with a convective condition on Gamma_1 converge, when
alpha\to\infty, to the corresponding system state, adjoint state, optimal
controls and optimal values that arise from the problem with a temperature
condition on Gamma_1. Also, we analyze the order of convergence in each case,
which turns out to be 1/alpha being new for these kind of elliptic optimal
control problems.
