Boundary Control For 1D Stefan Problem

The boundary control problem of the free boundary of a semilinear parabolic equation is studied in this paper:(?)Here f?C1(R × R),and Lipschitz continuousf(0,0)= 0.T>0,B>0,and 0<L*<L0<B.The iinitial condition are satisfied y0(x)?C2+?([0,T]),and the function satisfies(?).Free boundary conditions are satisfied L(t)?C1+?([0,T]),and meet the following conditions:0<L0?L(t)?B,t ?(0,T),(0.6)L'(t)=-yx(L(t),t),t?(0,T),(0.7)Where QL = {(x,t)|x ?(0,L(t)),t ?(0,T)} is the system space,y = y(x,t)is the system state,u(t)is the control function.The control function is applied to the whole system through the left boundary of the system.The paper is divided into five parts.The first part is the introduction,introduces the research background and the general idea of this paper.The second part is the prelimi-nary knowledge.The third part is the estimation towards Carleman.The fourth part is the observable inequality of the dual equation of the system.The fifth part is to obtain zero controllability of free boundary by fixed point theorem.The main conclusion is as follows:Assume that f ?C1(R×R),f' is Lipschitz continuous and f(0,0)= 0.Also,assume that T>0,B>0,and 0<L*<L0<B,Then(0.5)-(0.7)is locally null-controllable.More precisely,there exists ?>0,such that if ?y0?c2+?([0,L0])??,then y(x,T)=0,x?(0,L(T)),(0.8)...