We consider a steady-state heat conduction problem P-alpha with mixed boundary conditions for the Poisson equation depending on a positive parameter alpha, which represents the heat transfer coefficient on a portion Gamma(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 alpha. We prove that the optimal control g(opalpha). and its corresponding system u(gopalphaalpha) and adjoint pg(opalphaalpha) states for each alpha are strongly convergent to g(op), u(gop) and p(gop), 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 Gamma(1). We use the fixed point and elliptic variational inequality theories.