Inverse Scattering and Tomography, Delay Differential Equations in Cyber-Physical Systems.pdf (6.72 MB)

# 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.

In the final project, we consider a perturbed biharmonic operator and study the inverse scattering problem for this operator by investigating the recovery process of the magnetic field A and the potential field V. Using the high-frequency asymptotic of the scattering amplitude of the biharmonic operator, we prove the unique recovery of curl A and V − 1/2 ∇·A. By investigating the near-field scattering, we show that the high-frequency asymptotic expansion up to an error O(λ^ −4 ) (where λ is the frequency or the spectral parameter) recovers the same above quantities but does not provide any additional information about the magnetic and the potential fields. We also establish stability estimates for curl A and V −1/2 ∇·A.

Our engineering work is in the area of Cyber-Physical systems where we study Real-time hybrid simulation (RTHS). By the use of RTHS, which is an efficient technique that investigate cyber-physical system in a cost-effective way, we introduce powerful indicators to examine the structural behavior and seismic resilience of a structure through various settings.

## Funding

### NSF Grant DMS 1301646

### NSF Grant DMS 1600327

### NSF Grant DMS 1900475

### NSF Grant CNS 1136075

## History

## Degree Type

- Doctor of Philosophy

## Department

- Mathematics

## Campus location

- West Lafayette

## Advisor/Supervisor/Committee Chair

PLAMEN D. STEFANOV## Additional Committee Member 2

ANTONIO SA BARRETO## Additional Committee Member 3

KIRIL DATCHEV## Additional Committee Member 4

SHIRLEY J. DYKE## Usage metrics

## Categories

- Applied mathematics not elsewhere classified
- Calculus of variations, mathematical aspects of systems theory and control theory
- Mathematical physics not elsewhere classified
- Partial differential equations
- Pure mathematics not elsewhere classified
- Theoretical and applied mechanics
- Lie groups, harmonic and Fourier analysis
- Integrable systems (classical and quantum)
- Dynamical systems in applications
- Earthquake engineering
- Dynamics, vibration and vibration control
- Mechanical engineering not elsewhere classified

## Keywords

Light-Ray TransformsMinkowski Time-spacesHelgason’s type Support TheoremMicrolocal AnalysisFourier integral OperatorsFourier analysisPseudo-differential OperatorsAnalytic Wave front setStationary Phase MethodAnalytic ContinuationPartial Data, Dirichlet-to-Neumann mapRadon transformIntegral geometryBolker ConditionInverse ScatteringPerturbed Biharmonic operatorStability EstimatesReal-time Hybrid SimulationPredictive Stability indicatorRTHSPSIcontrol structure interactionApplied Mathematics not elsewhere classifiedCalculus of Variations, Systems Theory and Control TheoryMathematical Physics not elsewhere classifiedPartial Differential EquationsPure Mathematics not elsewhere classifiedTheoretical and Applied MechanicsLie Groups, Harmonic and Fourier AnalysisIntegrable Systems (Classical and Quantum)Dynamical Systems in ApplicationsEarthquake EngineeringDynamics, Vibration and Vibration ControlMechanical Engineering