Optimal Control Of One-Dimensional Semilinear Stefan Problem

In this paper,we consider the optimal control problem of the following one-dimensional Semilinear parabolic equations with free boundary:and the Stefan conditions:where QL={(x,t)|x ∈(0,L(t)),t ∈(0,T)},L(·)is the free boundary,y=y(·,·)is the state of the system,v=v(·,·)is a control,ω=(a,b)is a nonempty open set,0<a<b<L*<L0<B,1ω(·)denotes the characteristic function of the set ω,T>0 is given,f:R(?)R is a global Lipschitz continuous function,y0(·)is known.The cost functional is as follows:In this paper,the following optimal control problems are considered:select the control v(·,·),so that Under certain conditions,we have proved that there exists an optimal control for the above optimal control problems for small initial values.