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

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.