Optimal Control Of Two-Dimensional Stefan Problem

In this paper,we consider the optimal control problem of the Stefan problem:(?)(0-1)ρ(φ,0)=ρ0(φ),φ∈[0,2π],(0-2)(V-νΔV)·n=-?u/?n,(x,t)∈Σρ,(0-3)whereρ(φ+2π,t)=ρ(φ,t),(φ,t)∈R×[0,T],Ωρ(t)(?)｛x∈R2:x=(x1,x2)=r(cosφ,sinφ),0≤r<ρ(φ,t),φ∈[0,2π)},Ω0(?){x∈R2:x=(x1,x2)=r(cosφ,sinφ),0≤r<ρ0(φ),φ∈[0,2π)},Qρ(?)｛(x,t):x∈Ωρ(t),t∈(0,T)},Σρ(?)｛(x,t):x∈?Ωρ(t),t∈(0,T)}.T>0.V is the speed of the free boundary ?Ωρ(t).ν is a positive constant,u=u(x,t)is the state of the system,v=v(x,t)is the control function,which acts on the system through a non-empty open set ω.u0(x)and ρ0(φ)are given.(0-3)is the Stokes type condition satisfied by the free boundary.The cost functional is given as J(v(·))=∫Ωρ(T)u2(x,T)dx+?ω×(0,T)v2(x,t)dxdt.(0-4)We consider the follow optimal control problem,i.e.,find control v(·,·),such thatBy using the fixed boundary method and the Schauder fixed point theorem,we prove that under certain conditions,the above optimal control problem has optimal control for sufficiently small initial values.