Insensitizing Controls of a 1D Stefan Problem for the Semilinear Heat Equation

This paper deals with the existence of insensitizing controls for a 1D free-boundary problem of the Stefan kind for a semilinear PDE. The insensitizing problem consists in finding a control function such that some energy functional of the system is locally insensitive to a perturbation of the initial data. As usual, this problem can be reduced to a nonstandard null controllability problem of some nonlinear coupled system governed by a semilinear parabolic equation with a free-boundary and a linear parabolic equation. Nevertheless, in order to solve the later Stefan problem by the fixed point technique, we need to establish the null controllability of the linear coupled system in a non-cylindrical domain. An observability estimate for the corresponding coupled system in a non-cylindrical domain is established, whose proof relies on a new global Carleman estimate.