Stefan problem has been known as a thermodynamic model of liquid-solid phase transition since Josef Stefan's work in 1891. The Stefan problem is formulated by a diffusion partial differential equation defined on a time-varying spacial domain which obeys an ordinally differential equation driven by Neumann boundary value at the boundary. The state-dependency of the boundary makes the implicitly given dynamics of the boundary which is called "free boundary". While a numerical analysis of the Stefan problem has been considered intensely, the control related problems have been studied relatively fewer. The central goal of my theoretical work is to develop a boundary control design for the Stefan problem (more generally "free boundary PDEs") via "backstepping design" which has been intensively advanced in the recent decades.