The Optimal Control For The Free Boundary Of Two-phase Stefan Problem With Second Boundary Value Conditions

We investigate in this article the optimal control problem for the two-phase Stefan problem where Xi=kipi-1Ci-1,i=1,2,represent the diffusivities；ki,i=1,2,the conductivities；pi ,i=1,2,the densities；ci,i=1,2,the heat capacities;Ki=kipi-1L-1,i=1,2,L is the latent heat.All of the preceding constants are positive.0<b<1,f(·)≤0,9(·)≤0,φ(·)≥ 0,φ(·)≤0,T>0.Consider the cost functional s0(·)is a given position function,s(·;f,g)is the free boundary of the two-phase Stefan problem with respect to f(·)and g(·).f(·)and g(·)are control functions.The purpose of the control problem is expected to let the free boundary track the given location.We should consider two aspects in this article:One is the existence of the optimal control,employing the weak *convergence of the minimizing sequence and Mazur theorem； The other is to give the necessary condition for the existence of the optimal control by the limearization method.