Infinite-dimensional input-to-state stability and Orlicz spaces

TL;DR

This paper explores the relationship between input-to-state stability and integral input-to-state stability in infinite-dimensional linear systems, characterizing stability in terms of Orlicz spaces and analyzing specific cases like parabolic diagonal systems.

Contribution

It establishes a connection between input-to-state stability and Orlicz space-based stability, extending the understanding of stability in infinite-dimensional systems with unbounded control operators.

Findings

Integral input-to-state stability characterized via Orlicz spaces.

For parabolic diagonal systems, $L^inity$ stability notions are equivalent.

Results relate stability to admissibility in linear systems.

Abstract

In this work, the relation between input-to-state stability and integral input-to-state stability is studied for linear infinite-dimensional systems with an unbounded control operator. Although a special focus is laid on the case $L^{\infty}$, general function spaces are considered for the inputs. We show that integral input-to-state stability can be characterized in terms of input-to-state stability with respect to Orlicz spaces. Since we consider linear systems, the results can also be formulated in terms of admissibility. For parabolic diagonal systems with scalar inputs, both stability notions with respect to $L^\infty$ are equivalent.