  As opposed to the distributed control of parabolic PDE's, very few
contributions currently exist pertaining to the Dirichlet boundary condition
control for parabolic PDE's. This motivates our interest in the Dirichlet
boundary condition control for the phase field model describing the
solidification of a pure substance from a supercooled melt. In particular, our
aim is to control the time evolution of the temperature field on the boundary
of the computational domain in order to achieve the prescribed shape of the
crystal at the given time. To obtain efficient means of computing the gradient
of the cost functional, we derive the adjoint problem formally. The gradient is
then used to perform gradient descent. The viability of the proposed
optimization method is supported by several numerical experiments performed in
one and two spatial dimensions. Among other things, these experiments show that
a linear reaction term in the phase field equation proves to be insufficient in
certain scenarios and so an alternative reaction term is considered to improve
the models behavior.
