Carleman Inequality Of Backward Linear Parabolic Equations On Non-cylindrical Domains And Null Controllability Of Boundary For 1D Stefan Problem

In this paper,the boundary controllability of Stefan problem for one-dimensional semi-linear parabolic equation is studied:L'(t)=-a(x,t)yx(L(t),t),t ?(0,T).(0.2)where f(·,·)? C2(R × R)and f(·,·)is Lipschitz continuous,f(0,0)=0.Assuming given T>0,B>0,L0>0,a,(·,·)? W2,?((0,B)×(0,T)),a(x,t)?a>0,0<L*<L0<B,g(·,·)? C?,?/2((0,B)×(0,T)).y0(·)? C2+?([0,L0]),free boundary x=L(t)? C1+?([0,T]),and:0<L0?L(t)?B,t ?(0,T).(0.3)QL={(x,t)|x?(0,L(t)),t ?(0,T)}.y=y(x,t)is a state function of the system.u(·)is the control function.The control function acts on the whole system through the left boundary of the system.This paper is divided into five parts.In the first part,an introduction for the research background and a general research idea of this paper is given.In the second part,some preparatory knowledge applied in this paper are listed.In the third part,the Carleman inequality for the dual equation of linear parabolic equation on non-cylindrical domain is established.In the fourth part,we prove the zero controllability of linear parabolic equations on non-cylindrical domains.In the fifth part,zero controllability of free boundary is obtained by using fixed point theorem.The main result of this paper is as follows:If f(·,·)E C2(R × R),and Lipschitz continuous,f(0,0)=0,T>0,B>0 are given,0<L*<L0<B,Then(0.1)-(0.2)is locally zero controllable,i.e.there exists ?>0,such that if ?y0?C2+?([0,L0])??,exists u(·)?C?,?/2,such that y(x,T)=0,x ?(0,L(T)).(0.4)...