Exact Controllability,Backward Problems And Carleman Estimates For Coupled Systems Of Partial Differential Equations

This Ph.D dissertation mainly studies three main contents:Firstly,we study the ob-servability inequality and control properties for second-order hyperbolic systems with variable coefficients in inhomogeneous and anisotropic cases in n dimensions.By the classical Hilbert Uniqueness Method,we shall establish an observability inequality for the second-order hyperbolic systems with variable coefficients of high dimensions and apply it to obtain the exact controllability of the corresponding systems.The same re-sults for linear elastodynamic systems with variable coefficients in inhomogeneous and anisotropic media are provided to illustrate the application of the results.Secondly,we study the conditional stability for the backward problem of a principal strongly cou-pled hyperbolic-parabolic systems in a bounded domain in high dimensions.We will establish a Carleman estimate which is suitable for the backward problems and apply it to the backward problem of the principal strongly coupled hyperbolic-parabolic sys-tems to obtain the conditional stability.Thirdly,we study Carleman estimates for the inhomogeneous principal strongly coupled hyperbolic-parabolic systems in a bounded domain in n dimensions.This dissertation is divided into five chapters.In chapter Ⅰ,we introduce the research background of this dissertation.In chapter Ⅱ,we give some terminologies and some known basic results.In chapter Ⅲ,we study the exact controllability of the strongly coupled hyper-bolic systems with variable coefficients of high dimensions.In the first section,in order to completely state the results in this Chapter,we introduce the well-posedness of the strongly coupled hyperbolic systems with variable coefficients of high dimen-sions.We discuss the existence of solutions for initial/boundary value problems of the strongly coupled hyperbolic systems with variable coefficients of high dimensions by semigroups theory and Hille-Yosida Theorem.Then we give the definition of the energy of weak solutions and give a proof of the energy conservation for the initial/boundary value problems of the systems under consideration which means the uniqueness of the solution.In the second section,by using the common method of proving observability inequalities in the controllability theory of the evolution equations,we prove an observ-ability inequality of the strongly coupled hyperbolic systems with variable coefficients of high dimensions.In the third section,we introduce the definition of solution by transposition of the strongly coupled hyperbolic systems with variable coefficients of high dimensions and the definition of exact controllability.Then,by the observability inequalities obtained in the second section and classical Hilbert Uniqueness Method,we prove that,under the suitable assumptions,the strongly coupled hyperbolic systems with variable coefficients of high dimensions are exact controllable.At the end of this section,we consider the applications of these results to the observability inequalities and exact controllability for a single hyperbolic equation with variable coefficients of high dimensions.In the fourth section,in order to illustrate the application of the re-sults in the second and third sections,we consider the observability inequalities and exact controllability of the linear elastodynamic systems with variable coefficients in inhomogeneous and anisotropic media.In chapter Ⅳ,we study the conditional stability estimates for the backward prob-lem of the principal strongly coupled hyperbolic-parabolic systems.In the first sec-tion,we establish a Carleman estimate for the principal strongly coupled hyperbolic-parabolic systems which is suitable for the backward problems.In the second section,we prove the energy estimate about the principal strongly coupled hyperbolic-parabolic systems with initial/boundary values.In the third section,we apply the Carleman esti-mate proved in the first section to the backward principal strongly coupled hyperbolic-parabolic systems to obtain the conditional stability.Then,we optimize the conditional stability by the energy estimate obtained in the second section.In chapter Ⅴ,we study Carleman estimates for the inhomogeneous principal strongly coupled hyperbolic-parabolic systems.In this chapter,we assume that the coefficients in the systems under consideration satisfy suitable conditions,taking the weight func-tion e2sχ where φ(x,t)= eλφ(x,t)and φ(x,t)= |x-x0|2-β(t-T/2)2 for(x,t)∈Ω×(0,T),Ω(?)Rn is a bounded domain with C3-boundary.The main tool is integration by parts.