Date of Award
Doctor of Philosophy (PhD)
In this thesis, we will study the interaction between problems in control theory for partial differential equations and inequalities of the uncertainty principle type. The main results will concern the boundary observability of the viscoelastic wave equation and energy decay rates of damped wave equations. In the boundary case, we will prove what may be viewed as a higher dimensional version of Ingham's inequality, replacing the complex exponentials with Laplacian eigenfunctions.
For energy decay rates on the real line, we will use a version of the Paneah-Logvinenko-Sereda theorem for functions with Fourier support contained in multiple intervals. We prove the exact variation which we need and apply it to internal observability as well as decay rates for damped wave equations as well. We also give partial results in higher dimensions and some open problems.
We will also investigate the connection between compactness of localization operators and uncertainty principles from an abstract harmonic analysis perspective. We give some general results which are applied to the wavelet transform.
Green, Andrew Walton, "The Uncertainty Principle in Control Theory for Partial Differential Equations" (2020). All Dissertations. 2636.