This thesis is mainly devoted to a study of the exact controllability of semilinear dis-tributed parameter system. Some related problems, for example the observability, uniquecontinuation property and so on of semilinear and/or linear distributed parameter system,are also considered in this thesis.
This thesis is divided into three chapters.
In Chapter 1, under some assumptions which are di?erent from that of CarmichaelQuinn, LasieckaTriggiani and Seidman , we obtain several abstract results concerningthe exact controllability of the semilinear distributed parameter system, i.e., we prove thenull local exact controllability of semilinear system of first order by means of contrac-tion mapping principle (in this case we do not assume any compactness); we derive theglobal and/or local exact controllability of semilinear system of second order by means ofSchauder’s fixed point theorem (in this case we assume only the embedding of the relatedspaces having some compactness, which is reasonable for many concrete problems). Ourabstract result shows that: The observability of the dual system of the linearized systemimplies the exact controllability of the original semilinear system. Such a result is onlyproved for the linear system in the previous publications.
In Chapter 2, we mainly consider the exact boundary and internal controllability ofthe semilinear wave equations in n space dimension (n≥1) when the control are only act-ing on a part of the boundary or in a boundary layer with respect to a part of the boundary,compared to Zuazua’s results, who considered only the exact boundary controllability ofthe semilinear wave equation using control distributed on the whole boundary and theexact internal controllability of the semilinear wave equation in one space dimension. Byour abstract results in Chapter 1, we need to derive the observability inequality of the dualsystem of its linearized system, which is a linear wave equation with time-variant lowerorder term. In the literature, one can find two important methods to derive such sortof inequalities. The first is“the Multiplier Method+Compactness-uniqueness Argument”;The second is“Carleman-type Estimate+Compactness-uniqueness Argument”. We note