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.

翻译：这项工作提出了玻璃圆顶空气加热过程模型的衍生结果,在这个模型中,室内游泳池位于圆顶底部;问题可以从三维缩小到二维;主要目标是制定适当的优化问题,以便在特定时间后计算最佳空气加热;为此,制定了部分差分方程的加热过程模型,以及按时间要求部分差分方程式处理的优化问题;这样就可以使玻璃圆顶下的空气得到最佳加热,以便在特定时间之后达到理想的温度分布;讨论了最佳化问题的单独配方和适当的数字方法,即预测梯度方法;最后,提出了数字实验,表明最佳控制问题的实际表现及其讨论的数字解决办法方法。