Local Null Controllability Of A Free-boundary Problem For The Semilinear 1D Heat Equation

We investigate in this paper locall null controllability of a free boundary problem for the semilinear 1D heat equation:T>0,0<a<b<L_*<L₀,the initial y0∈C^2+α（[0,L₀]）are given,exist v∈C^α,α/2（?）,Find L∈C^1+α[0,T])（α∈（0,1/2））with 0<L_*≤ L（t）≤ B t ∈（0,T）（0.4）Consider the equation as follows:with the free boundary condition L’（t）=-yx（L（t）,t）,t∈（0,T）.（0.6）Here QL = {（x,t）:x ∈（0,L（t））,t ∈（0,T）}.y =y（x,t）is the state,v = v（x,t）is a control;it acts on the system at any time through the nonempty open set ω=（a,b）,0<a<b<L_* I_ω denotes the characteristic function of the set ω.The main method of this paper is to use locall null controllability results for the linear heat equation in a non-cylindrical domain and the regularity property of the solution,infer to carleman estimate of corresponding dual system and an observability inequality,finally find free-boundary and triplets by using fixed point theorem,thus come tue that the system is locall null controllability at time t = T.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 satisfy 0<ab<L_*<L₀<B.Then（0.4）-（0.6）is locally null-controllable.More precisely,there exists ε>0,such that if ‖y⁰‖C^2+α（[0,L₀]）≤ε,then y（x,T）=0 x∈（0,L（T））.