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

We investigate in this thesis the optimal control problem for the two-phase stefan prob-lem where for i=1,2, ki=kipi-1ci-1 represent the diffusivities ki the conductivities; pi the densities; ci the heat capacities; Ki= kipi-1L-1, L is the latent heat. All of the preceding constants are positive. T> 0,0< b< 1,f(·)> 0,g(·)> 0,φ(·)≥0,φ(·)≤0.Consider the cost functional where so(t) is a given position function, s(t;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 linearization method.