Approximate Controllability For Linearized Compressible Navier-Stokes System

In this paper,we study the approximate controllability of the initial boundary value problems of linearized compressible Navier-Stokes equations,where the control functions act on the mass equation and the momentum equation respectively.Since the approximate controllability of the original problem is equivalent to the unique continuation of the dual problem,the main research content of this paper is to prove the unique continuation of the dual problem by Hormander(5)(5)theorems.In the first chapter of this paper,we mainly introduce the background and significance of this topic,some relevant research status at home and abroad,the main research content and the structure of this paper.In the second chapter of this paper,we study the approximate controllability of the initial boundary value problem of linearized compressible Navier-Stokes equations on a one-dimensional bounded interval,where there is only one control function and it acts on the mass equation.In this paper,we first prove the adaptability of linearized systems,then prove the unique continuation of dual problems by using Hormander(5)(5)theorems,and finally obtain the approximate controllability of one-dimensional linearized compressible Navier-Stokes equations by using the unique continuation of dual problems.In the third chapter of this paper,we study the approximate controllability of the initial boundary value problems of linearized compressible Navier-Stokes equations over three dimensional bounded domains,where the control functions only act on the momentum equations.We adopt the similar proof method as the second chapter.The main research content is to prove the unique continuation of dual problems by Hormander(5)(5)theorems.Since this chapter deals with the three-dimensional case,it is more complicated than the second chapter to construct differential operators satisfying the conditions of the Hormander(5)(5)theorems.In the fourth chapter of this paper,we mainly introduce the research results of this paper,the difficulties of the research methods and the innovation of the research content.