In order to design a road de-icing device by heating, we consider in the one dimensional setting the optimal control of a parabolic equation with a non linear boundary condition of the Stefan-Boltzmann type. Both the punctual control and the corresponding state are subjected to a unilateral constraint. This control problem models the heating of a road during a winter period to keep the road surface temperature above a given threshold. The one-dimensional modeling used in this work is a first step of the modeling of a road heating device through the circulation of a coolant in a porous layer of the road. We first prove, under realistic physical assumptions, the well-posedness of the direct problem and the optimal control problem. We then perform some numerical experiments using real data obtained from experimental measurements. This models and the corresponding numerical results allow to quantify the minimal energy to be provided to keep the road surface without frost or snow.