Well-posedness And Regularity Of Control Systems For Partial Differential Equations

The well-posedness and regularity of control systems for partial differential equations is an important research topic in the distributed parameter systems control theory, which has important theoretical significance and application value. In this paper, we mainly study the well-posedness and regularity of Euler-Bernoulli plate equation and fourth order Schrodinger equation. The paper consists of four chapters.In Chapter 1, some research background, the research advance of the related work are given. Moreover, the main results obtained in this thesis are listed.Chapter 2 is devoted to the study on the well-posedness and regularity of Euler-Bernoulli plate equation. In Section 1 of Chapter 2, we consider the regularity of Euler-Bernoulli plate equation with constant coefficient and with hinged and clamped boundary by Dirichlet boundary control and collocated observation, respectively, and give the cor-responding feedthrough operator, thus we prove the systems are regular. In Section 2 of Chapter 2, we make use of Riemannian geometric methods to generalize the well-posedness and regularity for Euler-Bernoulli plate equation with hinged and clamped boundary by Dirichlet boundary control and collocated observation, respectively, to the cases where the coefficients are spatial variable dependent.Chapter 3 is devoted to the the study on the well-posedness and exact controllability of fourth order Schrodinger equation with clamped boundary. In Section 1 of Chapter 3, we discuss the well-posedness of fourth order Schrodinger equation with clamped bound-ary by Neumann boundary control and collocated observation. We show that the system is well-posed by proving the stability of the input and output. From this fact, we can learn that the corresponding closed-loop system under proportional output feedback control is exponentially stable. Moreover, the system is regular in the sense of G. Weiss, and the feedthrough operator is zero. Next, in Section 2 of Chapter 3, the well-posedness result is generalized to the clamped boundary by Dirichlet boundary control and collocated ob-servation. Similarly, we can learn that this system is regular in the sense of G. Weiss, and its feedthrough operator is zero. In Section 3 of Chapter 3, we prove fourth order Schrodinger equation with clamped boundary by the Dirichlet boundary control and col-located observation is exact controllable by establishing the observability inequality of the dual system. So, from the result of well-posedness, we can learn that the corresponding closed-loop system under proportional output feedback control is exponentially stable.Chapter 4 is devoted to the well-posedness and exact controllability of fourth order Schrodinger equation with hinged boundary. In this Chapter, Section 1 and 2 are con-cerned with the well-posedness of fourth order Schrodinger equation with hinged boundary by either Dirichlet or moment boundary control and collocated observation, respectively, we prove the systems are well-posed by prove the stability of the input and output. In addition, it is shown that the two systems are regular, and the feedthrough operators are zero. Moreover, in Section 3 of Chapter 4, we study the exact controllability of the two systems by establishing the observability inequalities of the dual systems, respectively. So, from this fact, it can obtain that the corresponding closed-loop systems under proportional output feedback control are exponentially stable.