Controllability Of Boundary For 1D Stefan Problem

In this paper,the boundary controllability of Stefan problem for one-dimensional semi-linear parabolic equation is mainly studied:(?)Assuming given T? 0,a(x,t)? W2,?((0,B)×(0,T)),a(x,t)? M,M is a positive number,ax,(0,t)=0,and 0?L*?L0 ?B.Arbitrary initial value is satisfied y0(x)?H01([0,L0]),g?L2((0,B)x(0,T)),Stefan boundary condition is satisfied L(t)?C1([0,T]),and:0<L0?L(t)?B,t ?(0,T).(0.6)L'(t)?-y(L(t),t),t?(0,T).(0.7)Where QL={(X,t)|x?(0,L(t)),t?(0,T)} is the overall system space.y=y(x,t)is a state function of the system.u(t)is the control function.The main structure of this paper is divided into five parts.The first part is the intro-duction which mainly introduces the research background,and some research results of the Stefan problem of parabolic equation.The second part lists the lemma,definitions.The third part mainly derives the dual equation and its global Carleman estimation.The fourth part derives the energy-related inequality that the solution of the dual equation satisfies through Carleman estimation and then proves the zero controllability of linear parabolic equations in non-cylindrical domains.In the fifth part,the local zero controllability of the boundary control of Stefan problem is proved by using fixed point theorem.One of the main conclusions of this paper is following:If f?c1(R),f is Lipschitz continuous and f(0)=0,given T? 0,B? 0,0 ?L*?L0?B,Then(0.5)-(0.7)is locally zero controllable.There exists ?? 0,such that if||y0||h01([0,L0])??,then:y(x,T)=0,x?(0,L(T)).(0.8)...