Weighted operator-valued function spaces applied to the stability of delay systems

TL;DR

This paper develops a framework using weighted operator-valued function spaces to analyze the stability of delay systems, extending classical control methods to more complex, infinite-dimensional settings.

Contribution

It extends Zen space theory to Hilbert-space valued functions and applies it to delay systems, linking $H^$ control techniques with operator-valued transfer functions.

Findings

Extended Zen spaces to Hilbert-space valued functions.

Analyzed delay systems with operator-valued transfer functions.

Connected $H^$ structure dependence to delays using extended Walton--Marshall technique.

Abstract

This paper extends the theory of Zen spaces (weighted Hardy/Berg\-man spaces on the right-hand half-plane) to the Hilbert-space valued case, and describes the multipliers on them; it is shown that the methods of $H^\infty$ control can therefore be extended to a family of weighted $L^2$ input and output spaces. Next, the particular case of retarded delay systems with operator-valued transfer functions is analysed, and the dependence of $H^\infty$ structure on the delay is determined by developing an extension of the Walton--Marshall technique used in the scalar case. The method is illustrated with examples.