Doctor of Philosophy
School of Mathematics and Applied Sciences
Liu, Wei-Jiu, Exact controlability for some partial differential equations of evolutional type, Doctor of Philosophy thesis, School of Mathematics and Applied Sciences, University of Wollongong, 1997. https://ro.uow.edu.au/theses/2039
In this thesis, we study the problem of exact controllabiHty for various partial differential equations of evolutional type. In Chapter 2, using Lions' Hilbert Uniqueness Method, we prove the wave equation with timedependent coefficients is exactly controllable with Neumann boundary controls. In Chapter 3, we first obtain the uniform stabihzation of the transmission wave equation with the aid of the classical energy method and multiplier techniques. Then, via Russell's "Controllability via Stabihzability" Principle, the exact controllability with Neumann or Robin boundary controls is derived. In Chapter 4, by estabhshing some uniqueness theorems for the transmission plate equation with lower-order terms, we prove that the equation is exactly controllable. In Chapter 5, the uniform boundary feedback stabilization of energy in the higher-dimensional linear thermoelasticity is established. Then, via Russell's "Controllability via Stabilizability" Principle, the partial exact controllability with boundary controls is obtained without smallness restrictions on coupling parameters. In Chapter 6, by making use of Schauder's fixed point theorem, the inverse function theorem and Browder-Minty's surjective theorem, we prove that nonlinear plate equations are globally (or locally) exactly controllable with Dirichlet boundary controls under various assumptions on the nonlinear terms. Finally, in Chapter 7, using Browder-Minty's surjective theorem, we show that the semilinear heat equation with distributed controls is exactly controllable if the nonlinearity is globally Lipschitz continuous.