Operator-valued multiplier theorems for causal translation-invariant operators with applications to control theoretic input-output stability

Christopher Guiver, Hartmut Logemann, Mark Opmeer

Research output: Contribution to journalArticlepeer-review

Abstract

We prove an operator-valued Laplace multiplier theorem for causal translation-invariant linear
operators which provides a characterization of continuity from H^\alpha(R,U) to H^\beta(R,U (fractional U-valued Sobolev spaces, U a complex Hilbert space) in terms of a certain boundedness property of the transfer function (or symbol), an operator-valued holomorphic function on the right-half of the complex plane. We identify sufficient conditions under which this boundedness property is equivalent to a similar property of the boundary function of the transfer function. Under the assumption that U is separable, the Laplace multiplier theorem is used to derive a Fourier multiplier theorem. We provide an application to mathematical control theory, by developing a novel input-output stability framework for a large class of causal translation-invariant linear operators which refines existing input-output stability theories. Furthermore, we show how our work is linked to the theory of well-posed linear systems and to results on polynomial stability of operator semigroups. Several examples are discussed in some detail.
Original languageEnglish
JournalMathematics of Control, Signals, and Systems
Early online date30 May 2024
DOIs
Publication statusE-pub ahead of print - 30 May 2024

Data Availability Statement

No datasets were generated or analysed during the current study.

Keywords

  • Causal translation-invariant operators
  • Fourier transform
  • Mathematical systems and control theory
  • Operator-valued multipliers

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