We study the asymptotic behavior of an optimal distributed control problem where the state is given by the heat equation with mixed boundary conditions. The parameter α intervenes in the Robin boundary condition and it represents the heat transfer coefficient on a portion
$\varGamma _{1}$
of the boundary of a given regular n-dimensional domain. For each α, the distributed parabolic control problem optimizes the internal energy g. It is proven that the optimal control
$\hat{g}_{\alpha}$
with optimal state
$u_{\hat{g}_{\alpha}\alpha}$
and optimal adjoint state
$p_{\hat{g}_{\alpha}\alpha}$
are convergent as α→∞ (in norm of a suitable Sobolev parabolic space) to
$\hat{g},$
$u_{\hat{g}}$
and
$p_{\hat{g}},$
respectively, where the limit problem has Dirichlet (instead of Robin) boundary conditions on
$\varGamma _{1}.$
The main techniques used are derived from the parabolic variational inequality theory.
