Browsing by Author "Haataja, Anna"

The purpose of this thesis is to look from two different perspectives how wave equations can be solved. These are the forward problem of wave equations as partial differential equations of the initial or boundary-value type, and later in the framework of control theory. First, to provide a meaningful solution space, the basics of the theory of weak derivatives and Sobolev spaces are discussed along with some approximation, extension, and embedding theorems. Then, the initial or boundary value problem of the wave equation is defined, its weak solutions are constructed based on general hyperbolic partial differential equations, and the existence and uniqueness of said solutions is proved. The last part of this thesis concentrates on linear control theory: controllability of a linear system, and especially how it can be defined and proven form the first half of the last chapter. The other half is reserved for wave equations in control theory and why it is possible to reduce a wave control problem to solving the control problem of the aforementioned linear system.