  A range of optimization cases of two-dimensional Stefan problems, solved
using a tracking-type cost-functional, is presented. A level set method is used
to capture the interface between the liquid and solid phases and an immersed
boundary (cut cell) method coupled with an implicit time-advancement scheme is
employed to solve the heat equation. A conservative implicit-explicit scheme is
then used for solving the level set transport equation. The resulting numerical
framework is validated with respect to existing analytical solutions of the
forward Stefan problem. An adjoint-based algorithm is then employed to
efficiently compute the gradient used in the optimisation algorithm (L-BFGS).
The algorithm follows a continuous adjoint framework, where adjoint equations
are formally derived using shape calculus and transport theorems. A wide range
of control objectives are presented, and the results show that using
parameterised boundary actuation leads to effective control strategies in order
to suppress interfacial instabilities or to maintain a desired crystal shape.
