Delay Systems with Variable Delay

Abstract

In this thesis, we investigate questions about the ways to analyse the stability of input-output systems with delays that may be variable. This investigation divides the thesis into six chapters of which four present ways to analyse the stability and they are linked to each other directly or indirectly. After detailing the necessary background in Chapter 1, the focus switches in Chapter 2 to analysing the Hinfty stability of the retarded ordinary delay system with constant delays and operator-valued transfer function. This is achieved by developing an extension of the Walton-Marshall technique [39, 47], originally presented in the purely scalar case, to matrices and some operator cases. Therefore, we start with the main result of the second chapter which is concerned with the transfer function in operator-valued Hinfty. Then we adapt their methods to study a system with bounded operators, which requires us to consider the spectrum of the operators. From this we have a complex version of the Walton-Marshall formula. Additionally, we have a simple result about subnormal operators. From analysing the Hinfty-stability of ordinary delay systems, we move in the third chapter to looking at different kinds of stability of variable delay systems. This can be approached by considering the stability of an ordinary delay system that is very close to the variable delay one in order to ensure similar properties under specific conditions. Because of this we looked at a particular paper of Bonnet and Partington [5] in order to make extensions to the BIBO stability and Hinfty-stability results covered in their paper and to consider a more general version of stability, namely L p stability for p from 1 to infinity. Additionally, we analyse the three main versions of stability for normal or subnormal operators. Next, by changing the variables, the variable delay system mentioned in the previous chapter can often be transformed into an ordinary system with weights and constant delays, which may enable us to analyse the stability of the varying delay system. The transformed equation can sometimes be solved and we discuss the nature of instabilities that make the output of the system fail to lie in L2 (or L infty) whereas its input is in L2 (or Linfty). The main result of the fifth chapter is a theorem that combines two results. The first one is due to Jacob, Partington and Pott [29], which will be seen as a scalar version of our results. It involves stability using weighted L2 spaces which correspond by the Laplace transform to Zen spaces on the half plane. However, we extend the theory of Zen spaces (weighted Hardy/Bergman spaces on the right-hand half-plane) to the Hilbert-space valued case. The second result that we are generalizing is Plancherel's Theorem [2, Thm. 1.8.2] which is for Hilbert-space valued functions, but with different spaces. Then we describe the multipliers on weighted Hardy/Bergman spaces on the right-hand half-plane. It is shown that the methods of infty control can therefore be extended to certain weighted L2 input and output spaces. Additionally, we focus on the BIBO stability and L2-stability of autonomous and non-autonomous systems without delay but with weights, g1 and g2, under suitable conditions on g1 and this is illustrated by special examples. Furthermore, we give some results on types of stability that link the output of the ordinary delay system with the output of the variable delay system. Finally, we give some suggestions for further research.