  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.
