Bohl-Perron type stability theorems for linear difference equations with infinite delay

TL;DR

This paper establishes a precise equivalence between exponential stability and $ ext{l}^p$-input $ ext{l}^q$-state stability for linear difference equations with infinite delay under certain boundedness conditions, extending stability theory.

Contribution

It proves a sharp criterion linking exponential stability and $ ext{l}^p$-input $ ext{l}^q$-state stability for equations with fading memory, independent of phase space parameters.

Findings

Equivalence holds for all $(p,q) eq (1, ext{infinity})$ under boundedness.

The criterion is sharp and phase space independent.

Similar criteria likely do not apply to non-fading memory spaces.

Abstract

Relation between two properties of linear difference equations with infinite delay is investigated: (i) exponential stability, (ii) $\l^p$-input $\l^q$-state stability (sometimes is called Perron's property). The latter means that solutions of the non-homogeneous equation with zero initial data belong to $\l^q$ when non-homogeneous terms are in $\l^p$. It is assumed that at each moment the prehistory (the sequence of preceding states) belongs to some weighted $\l^r$-space with an exponentially fading weight (the phase space). Our main result states that (i) $\Leftrightarrow$ (ii) whenever $(p,q) \neq (1,\infty)$ and a certain boundedness condition on coefficients is fulfilled. This condition is sharp and ensures that, to some extent, exponential and $\l^p$-input $\l^q$-state stabilities does not depend on the choice of a phase space and parameters $p$ and $q$, respectively. $\l^1$-input $\l^\infty$-state stability corresponds to uniform stability. We provide some evidence that similar criteria should not be expected for non-fading memory spaces.