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

Guiver, Chris; Logemann, Hartmut; Opmeer, Mark R.

Abstract

We prove an operator-valued Laplace multiplier theorem for causal translation-invariant linear operators which provides a characterization of continuity from Hα(R, U) to Hβ(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.

Citation

Guiver, C., Logemann, H., & Opmeer, M. R. (2024). Operator-valued multiplier theorems for causal translation-invariant operators with applications to control theoretic input-output stability. Mathematics of Control, Signals, and Systems, 36(4), 729-773. https://doi.org/10.1007/s00498-024-00387-4

• Journal Article Type: Article
• Acceptance Date: May 6, 2024
• Online Publication Date: May 30, 2024
• Publication Date: Dec 1, 2024
• Deposit Date: May 6, 2024
• Publicly Available Date: May 30, 2024
• Print ISSN: 0932-4194
• Electronic ISSN: 1435-568X
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 36
• Issue: 4
• Pages: 729-773
• DOI: https://doi.org/10.1007/s00498-024-00387-4
• Keywords: Causal, translation-invariant operators, Fourier transform, Mathematical systems and control theory, Operator-valued multipliers