This work presents the derivation of a model for the heating process of the air of a glass dome, where an indoor swimming pool is located in the bottom of the dome. The problem can be reduced from a three dimensional to a two dimensional one. The main goal is the formulation of a proper optimization problem for computing the optimal heating of the air after a given time. For that, the model of the heating process as a partial differential equation is formulated as well as the optimization problem subject to the time-dependent partial differential equation. This yields the optimal heating of the air under the glass dome such that the desired temperature distribution is attained after a given time. The discrete formulation of the optimization problem and a proper numerical method for it, the projected gradient method, are discussed. Finally, numerical experiments are presented which show the practical performance of the optimal control problem and its numerical solution method discussed.