  In this paper, a sub-optimal boundary control strategy for a free boundary
problem is investigated. The model is described by a non-smooth
convection-diffusion equation. The control problem is addressed by an
instantaneous strategy based on the characteristics method. The resulting time
independent control problems are formulated as function space optimization
problems with complementarity constraints. At each time step, the existence of
an optimal solution is proved and first-order optimality conditions with
regular Lagrange multipliers are derived for a penalized-regularized version.
The performance of the overall approach is illustrated by numerical examples.
