Controllability For Some Types Of Nonlinear Wave Equations

Controllability is one of the important problems studied in distributed parameter control theory. We study the controllability for the wave equations in this thesis.In Chapter 1, we introduce briefly a research summary of controllability of distributed parameter control theory and the main contents of this thesis.In Chapter 2, we study the exact controllability for the uncoupled system composed of several linear damped wave equations with different propagation speed. We prove the system can be controlled exactly by the same control (this property is called simultaneous exact controllability). Our method is to study the equivalence between the controllability of the original and the observability of the dual system by using the classic Hilbert uniqueness method (HUM). The simultaneous exact controllability for linear wave equations is proved by proving an observability inequality.In Chapter 3, the internal exact controllability for the nonlinear wave equation in one space dimension ytt-yxx+9(y)yt=h1?is studied , where g(y) is a nonnegative function, the damping term g(y)yt is called degen-erate and h is the internal control. For the case, (?)g(s)/ln|s|<?(? is a sufficiently smallpositive number), we obtain the global exact controllability for the equation with Dirichlet boundary condition. The proof is based on the linearization principle and explicit observ-ability estimates for a linear damped wave equation. For the case, g(s)=|s|,we only get a local exact controllability by means of the Banach fixed-point theorem.In Chapter 4, we consider two desensitizing control problems for the semilinear wave equations. First, assume that the nonlinearity f satisfying (?). We prove that there exists an control such that the Neumann boundary observation is insensitive to the perturbation from initial data. Secondly, assume that the nonlinearity f is global Lipschitz continuous. We prove that there exists an control such that the internal observation is approximately insensitive to the perturbation from initial data.