Carleman Estimates And Null Controllability Of A Class Of Degenerate Parabolic Equations

In this paper,we study the null controllability problem of the control system dominated by the degenerate parabolic equation where ?:=?-1,1?×?0,1?,?????,the control u is supported in the subset ?.k?0?= Osatisfies and To time T,if every f₀?L₂???,there exists u?L₂??0,T?×??such that the solution of equation above satisfies f?T,.,.?= 0,then the system is null controllable in T.System is null controllable if there exists T>0 such that the system is null controllable in time T.The conclusion we give in this paper is that for ???0,2?,the system is controllable in any time.In this paper,we use the equivalence of null controllability and observability to prove that null controllability is transformed into proving observable inequality.The Fourier method is used to reduce the dimension of the problem,which makes the problem easier and then give the estimate of dissipation speed of the one dimension problem.After that,we prove the Carleman inequality which is the most important of the article,and then the observable inequality is proved by the inequality.Finally,the proof of null controllability is completed.