Controllability Problem Of Terminal For The Free Boundary Of Two-Phase Stefan Problem

In this article, we investigate the controllability problem of terminal of the free boundary for the two-phase Stefan problem. where i=1,2, ?i2=?i-1c-1 represent the diffusivities; ?i the densities; ci the heat ca-pacities;Ki=?i-1L-1, L is the latent heat. All of the preceding constants are positive. T>0,0<b<1,f(·)?0,g(·)?0,?(·)?0,?(·)?0.Let T>0,?(t)?0,?(t)?0,?0?(0,min{b,1-b}). For any x0?[?0,1-?0], we want to find the control functions f(t) and g(t), such that the free boundary s(t) of Stefan problem can reach x0 at time T, namely s(T)= x0.To solve this problem, we assume then considering the existence of u(x,t) in the left region D1 and the existence of v(x,t) in the left region D2, where we can not use the separation of variables to solve this problem, because the regions are not rectangular regions. Firstly, we draw on the foundings of wave equation's separation of variables in the rectangular region, under the some conditions, the Fourier series solutions of problem for determining solution are given in the two trapezoid regions; secondly, by using the Stefan conditions satified by the free boundary, that iswe investigate the requirement of f(t) and g(t), such that (u(x, t),v(x, t), s(t)) are so-lution of the Stefan problem. Furthermore, the free boundary s(t) can reach position x0 at time T. Namely, the free boundary s(t) of Stefan problem is controllable at time T under a given range about x0.The main conclusions are as follows:(?) Fourier type series solution of u(x,t) in the left region D1 and v(x,t) in the right region D2 are gicen under certain assumptions;(?) a sufficient condition of the terminal controllability of the free boundary s(t).