Properties of Delay Systems and Diffusive Systems

In this thesis, we investigate questions about the properties of delay systems and diffusive systems as well as Hankel and weighted Hankel operators. After detailing the necessary background in Chapter 1, in Chapter 2 the focus is on the development of methods to study the stability of delay and fractional systems. This analysis is carried forward using some BIBO and H∞ stability tests. Generalisation of the Walton-Marshall method [38] enable us to move from the single and multi-delay cases to fractional delay systems. This method gives procedures for finding stability windows as the delay varies. Chapter 3 is concerned with diffusive systems. Via convenient adaptations of some tests due to Howland [19], it becomes possible to give necessary and sufficient conditions for the Hankel operator and the weighted Hankel operator to be nuclear. Also, in this Chapter we introduce more general weighted Hankel operators and discuss their boundedness. Here the reproducing kernel test plays an essential role in testing boundedness. Some fundamental examples are given to support our work. In Chapter 4 here we investigate questions regarding approximating infinitedimensional linear system by finite-dimensional ones. Moreover, we develop more research on the rate of decay of singular values of the associated Hankel operator. In Chapter 5 we mainly focus on diffusive systems defined by holomorphic distributions and measures on a half plane. In particular we look at the nuclearity (trace class) and Hilbert-Schmidt properties of such systems. Moreover, we begin further study of explicit examples of weighted Hankel operators for which we did not know whether they were bounded, those examples already introduced in Chapter 3. In Chapter 6 the boundedness of weighted Hankel corresponding to diffusive systems is analysed using the theory of Carleson measures. Chapter 7 gives some suggestions for further work.