Local controllability of a free-boundary problem for a class of one-dimensional degenerate parabolic equations

This paper is devoted to a study of the controllability of a free-boundary problem for a class of one-dimensional degenerate parabolic equations with distributed controls, locally supported in space. We prove that for any $T>0$, if the initial state is sufficiently small, there exists a control that drives the state exactly to rest at time $t=T$. The proof is based on Schauder’s fixed point theorem, combined with appropriate estimates for solutions to degenerate parabolic equations and for control functions.