  This paper concerns the null controllability of the two-phase 1D Stefan
problem with distributed controls. This is a free-boundary problem that models
solidification or melting processes. In each phase, a parabolic equation,
completed with initial and boundary conditions must be satisfied; the phases
are separated by a phase-change interface where an additional free-boundary
condition is imposed. We assume that two localized sources of heating/cooling
controls act on the system (one in each phase). We prove a local null
controllability result: the temperatures and the interface can be respectively
steered to zero and to a prescribed location provided the initial data and
interface position are sufficiently close to the targets. The ingredients of
the proofs are a compactness-uniqueness argument (to deduce appropriate
observability estimates adapted to constraints) and a fixed-point formulation
and resolution of the controllability problem (to deduce the result for the
nonlinear system). We also prove a negative result corresponding to the case
where only one control acts on the system and the interface does not collapse
to the boundary.
