Finding solvable subsystems 

Eskil Rydhe will present his doctoral thesis in mathematics at Lund University in 2017. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Professor Jonathan Partington at the University of Leeds, UK. 

 
The goal of the project is to advance research in some of the classic subjects in operator theory, complex analysis, and harmonic analysis. Even though such research concerns problems in pure mathematics, it can also be applied to control theory. The main objective is how accurately the temporal evolution of a system can be described mathematically. 

 
A vibrating guitar string can be regarded as an example of a linear system. Given its initial form there is a rule, a function, which describes the shape of the string at any point in time. The rules that describe the evolution of such systems over time form a semigroup. Advancing the theory of such semigroups is one of the goals of the project. 

 
One problem that often arises in the mathematical description of linear systems is that some initial conditions of the system are inadmissible, meaning that even a small perturbation in the initial conditions may result in a large change to a later state of the system. It is then not possible to produce a meaningful mathematical description of how the system develops over time, because in practice all the values are known only to a certain degree of precision. Even though it is not possible to describe the time evolution of such systems, it is possible to make certain observations. 

 
These observations may depend only on a portion of the system. For example, the sound of a vibrating string consists of the fundamental tone and a sequence of overtones. Infinitely many overtones exist in the mathematical description of the sound, but in many circumstances only a finite portion of the information is necessary. For example, in the case of a vibrating string, one may choose to observe only sound intensity, or only the audible part of the sound. Information about the system can thus be reduced to a single key value. In some cases this process leads to that all reasonable initial conditions also become admissible. Developing a theory to determine which observations are admissible in this way and which are not is a further goal of the project.