The temperature distribution of a heated body can be described by a well-known parabolic Initial-boundary value problem. In this thesis, we consider some control-theoretic questions arising in connection with such a heating process. Suppose that we can vary (subject to certain restrictions) the temperature of the medium which surrounds the body. The task is to choose the temperature under the given restrictions in such a manner that the temperature distribution of the body at a time T comes as close as possible to some desired temperature. Assume we can reach to the mentioned desired temperature at time level T. We consider the optimal control of the above control problem. This problem consists of finding an admissible boundary control, to minimize an objective functional, which In general depends on time, positions of the points of the body and the control variable.