| Abstract: | The problem of controlling the temperature distribution in a solid cylinder whose length varies with time and with one end in contact with a constant temperature medium is considered. This problem is motivated from that of controlling the temperature and thermal gradient inside a crystal pulled from a melt by the Czochralski method. Boundary feedback controls are derived by considering the time rate of change of a cost functional involving the deviations of both the solid temperature and its gradient from their desired values. The derived feedback controls consist of spatially distributed proportional-plus-rate and lag compensators and a non-linear feedback control involving the temperature gradient at the cylinder surface and the velocity of the spatial domain boundary. The resulting feedback-controlled system has the property that the cost functional along any motion decreases monotonically to zero with time. A numerical scheme for solving the partial differential equation of the feedback-controlled system is proposed. Typical numerical results on the dynamic behaviour of the feedback-controlled system obtained by means of the proposed scheme are presented. |
