A dual notion to BIBO stability

TL;DR

This paper explores the relationship between BIBO stability and LILO stability in infinite-dimensional systems, showing their duality and equivalence in finite dimensions, and clarifying their properties in general cases.

Contribution

It establishes the duality between BIBO and LILO stability for infinite-dimensional systems and clarifies their equivalence in finite-dimensional cases.

Findings

BIBO and LILO stability are dual in infinite-dimensional systems.

In finite dimensions, BIBO and LILO stability are equivalent.

Duality between BIBO and LILO stability is preserved in general cases.

Abstract

In this paper we consider BIBO stability of infinite-dimensional linear state-space systems and the related notion of $L^1$-to-$L^1$ input-output stability (abbreviated LILO). We show that in the case of finite-dimensional input and output spaces, both are equivalent and preserved under duality transformations. In the general case, neither of these properties is satisfied, but BIBO and LILO stability remain dual to each other.