Approximate Controllability Of A Class Of Weakly Degenerate Parabolic Equations Under Boundary Control

In this paper,the author investigates the boundary control problem for a class of degenerate parabolic equations,in which the control function acts on the degenerate boundary and the boundary condition is the second boundary condition.At the same time,the approximate controllability of the problem is obtained.The results show that for any objective function,a control function can be found,so that the solution of the problem can be sufficiently close to the objective function in a finite time.In the first chapter,this paper introduces the background of this kind of problem,the relevant work at home and abroad,as well as the main problems,methods and results of the research.In the second chapter,this paper studies a class of linear degenerate parabolic equations with convection terms and proves the approximate controllability of these equations under boundary control.Previous studies on the controllability theory of degenerate parabolic equations with convection terms are related to the dependency of convection term and diffusion term.In the paper,the author considers the case that the convection term does not depend on diffusion term.The approximate controllability of the boundary control problem for degenerate parabolic equations is studied by establishing a new Carleman inequality,which is proved by the undetermined exponential method.In the third chapter,this paper discusses the approximate controllability of semilinear degenerate parabolic equations.As we know,the difficulty of proving the approximate controllability of nonlinear problems is the proper compactness estimates of the solutions of the linearized problem and its conjugate problem,which requires us to give more accurate compactness estimates.Firstly,the semilinear problem is linearized,and the well-posedness of linear problem is established,then prove the approximate controllability of the linear problem.Secondly,the necessary compactness estimates are made for the solution of the linear problem and its conjugate problem.Finally,it is proved that the semilinear problem is approximately controllable according to Kakutani fixed point theorem.In the fourth chapter,this paper summarizes the main results obtained in the paper,as well as the new methods and new ideas.