Local Zero Controllability Of Stefan's Problem In A One-dimensional Semilinear Parabolic Equation

In this paper,we consider the local zero controllability problem for the free boundary of the following one-dimensional semilinear parabolic equation:(?)L(0)=L0,(0.2)L'(t)=-a(x,t)yx(L(t),t),t ?[0,T],(0.3)where QL={(x,t)|x ?(0,L(t)),t?(0,T)}.y=y(x,t)is the state of the system,x=L(t)is the free boundary,v=v(x,t)is the control function,Through the non-space open set w=(b,c)applied to the system,1w represents the eigenfunction of the set w?T>0,L*>0 is given,and 0<b<c<?<L0<L*,p(x,t)?a>0,Let's assume that y0,g is given,that f is given,and that it has:0<??L(t)?L*,t?[0,T],(0.4)The aim of this paper is to consider the local controllability of the above system.In the study ??L(t)?L*,with f? C(R)and Lipschitz continuity,f(0)=0.The solutions of the system(0.1)-(0.3)are locally zero-controllable for small initial values at time T,i.e.there is an ?>0,for any y0(t)?C2+1/2([0,L0])have ?y0?C2+1/2([0,L0])??,there exists a control v(·,·)?C1/2,1/4(QL),such that y(x,T)=0,x?(0,L(T)).(0.5)...