Inverse Scattering and Tomography, Delay Differential Equations in Cyber-Physical Systems

This dissertation concludes three mathematical works in Inverse Problems as well as an engineering work in Cyber-Physical Systems. Our mathematical works are in the area of Inverse Problems and Scattering Theory where the main focuses are on Support theorem of Vectorial Light-ray transform in Minkowski Spaces, Dynamical X-ray Tomography, and Inverse Scattering of the biharmonic operator.

For the first project, we prove a support theorem for vector fields whose integral lines vanish on an open set of light-like lines. In this work, we employ the method of microlocal analysis and Pseudo Differential Operators. We illustrate the application of our results for the inverse recovery of the hyperbolic Dirichlet-to-Neumann map through various examples.

The second project is motivated by an inverse problem arising from medical imaging where we investigate a dynamic operator, A , integrating over a family of level curves when the object changes between the measurements. Microlocal analysis is used to determine which singularities can be recovered by the data-set. We prove that not all singularities can be recovered, depending on the particular movement of the object compare to the X ray source. We then find sufficient conditions under which the reconstruction is possible. We also show that one can establish stability estimates and injectivity results under the same conditions, i.e. the Visibility, the Local and Semi-global Bolker conditions. We illustrate the implementation of our results in Fan-beam geometry.