Local Null Controllability And The Existence Of Time Optimal Control Of A Free-boundary Problem For The Semilinear 1D Heat Equation

This paper mainly deals with the local null controllability and the existence of time optimal control of a free-boundary problem for the semilinear 1D variable coefficient heat equation.let 0<L*≤ L(t)≤ B,t∈(0,T)we will considerHere,T>0.B>0 is fixed,QL=｛=(x,t):x ∈(0,L(t)),t∈(0,T)};y = y(x,t)is the state and v = v(x,t)is a control;It acts on the system through the noempty open set ω=(a,b),with 0<a<b<L*;1ω denotes the characteristic function of the set ω;a∈ C1/2,(Q0),and a>0.For the controllability result,we proved that,and if the initial state is sufficiently"small",there exists a control that satisfies y(x,T)=0,x∈(x,;(T)).For the time optimal control problem:(P):min{T:y(x,T)=0,x∈(0,L(0)),v ∈ Vρ,y is the solution to the equation corresponding to v},We prove that if the initial state is sufficiently "small",there exists at least one time optimal control for problem P.