Neumann boundary optimal control problems governed by parabolic variational equalities

We consider a heat conduction problem $S$ with mixed boundary conditions in a $n$-dimensional domain $\Omega$ with regular boundary and a family of problems $S_{\alpha}$ with also mixed boundary conditions in $\Omega$, where $\alpha>0$ is the heat transfer coefficient on the portion of the boundary $\Gamma_{1}$. In relation to these state systems, we formulate Neumann boundary optimal control problems on the heat flux $q$ which is definite on the complementary portion $\Gamma_{2}$ of the boundary of $\Omega$. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system state and the adjoint state when the heat transfer coefficient $\alpha$ goes to infinity. Furthermore, we formulate particular boundary optimal control problems on a real parameter $\lambda$, in relation to the parabolic problems $S$ and $S_{\alpha}$ and to mixed elliptic problems $P$ and $P_{\alpha}$. We find a explicit form for the optimal controls, we prove monotony properties and we obtain convergence results when the parameter time goes to infinity.