The modeling of industrial crystal growth processes leads to an
extremly complicated coupled system of partial differential equations.
The attempt to develop a satisfactory theory of optimal control
problems for this system is certainly a mathematical "grand
challenge". Therefore simplifications are indicated, which still
capture the main features of the original problem. We give an overview
over a hierarchy of increasing complexity for such simplified optimal
control problems that have been studied in the past years by (actual
and former) members of WIAS and of the group of F. Troltzsch at the TU
Berlin.