Dynamical and Spectral Inverse Problem for the Wave Equation

In this thesis we derive and give methods to solve the Dynamical Inverse Problem and the Spectral Inverse Problem for the one dimensional wave equation. In these problems, we have a semi-infinite spatial axis with constant wave speed and an unknown potential for the system. Given information about the system may only pertain to the boundary of the spatial axis. In the Dynamical Inverse Problem, we recover the potential from the Response Operator which represents the boundary measurement. In the Spectral Inverse Problem, we recover the potential from the Spectral Data of the Schrödinger Operator. The solution methods to both problems rely on the exact controllability of the underlying wave equation.