Null Controllability Of Nonhomogeneous Degenerate Parabolic Equations

This paper mainly studies whether three kinds of nonhomogeneous degenerate parabolic equations have null controllability,including two kinds of single equations and one kind of coupled equations.Because the null controllability of the original problem is equivalent to the observability of its dual problem,the controllability problem is transformed into an observability problem by considering the dual problem of the original problem.After derivation,it can be proved that there is a control function that makes the solution of the original problem zero at the terminal time,that is,the equation has null controllability.The first chapter is the introduction,which first introduces the research background of the problem.Then we introduce the research content of this paper,the null controllability of nonhomogeneous degenerate parabolic equations.Finally,it introduces the structure of this paper.In Chapter 2,we prove the well-posedness of nonhomogeneous degenerate parabolic equations.Firstly,some symbolic are given,and then the well-posedness of the nonhomogeneous degenerate parabolic equation studied in this chapter is proved.In Chapter 3,we prove the null controllability of nonhomogeneous degenerate parabolic equations when 0<?<2.Firstly,the optimal control problem and two important inequalities,Carleman inequality and observable inequality,are introduced.Then it is proved that there is a control function to make the equation null controllable.In Chapter 4,we prove the null controllability of nonhomogeneous degenerate parabolic equations when 0<??1/2 or 5/4 ??<2.The proof method is similar to that in Chapter 3.Due to the change of ?,the Carleman inequality and nonhomogeneous term in this chapter change.Although the scope of ? in this chapter is smaller than that in Chapter 3,the scope of nonhomogeneous term is wider than that requied in Chapter 3.In Chapter 5,we prove the null controllability of a class of nonhomogeneous coupled degenerate parabolic equations.Firstly,we study the the well-posedness of the solution of a single equation and Carleman inequality are studied.The Carleman inequality of coupled equations is studied by using the Carleman inequality of single equation.Finally,the null controllability of the original problem is proved by deriving the observable inequality of the dual problem.The sixth chapter is the conclusion,which summarizes the main conclusions and innovations.