Convergence of Distributed Optimal Controls on the Internal Energy in Mixed Elliptic Problems when the Heat Transfer Coefficient Goes to Infinity

We consider a steady-state heat conduction problem P _α with mixed boundary conditions for the Poisson equation depending on a positive parameter α , which represents the heat transfer coefficient on a portion Γ ₁ of the boundary of a given bounded domain in R ⁿ . We formulate distributed optimal control problems over the internal energy g for each α . We prove that the optimal control g_ op _α and its corresponding system u_ g_ op _α α and adjoint p_ g_ op _α α states for each α are strongly convergent to g _op , u_ g _op and p _ g _op , 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 Γ ₁ . We use the fixed point and elliptic variational inequality theories.