Comparison Between Different Notions of Stability for Laurent Systems

In this article, we examine a particular family of infinite-dimensional discrete autonomous systems given by a first-order state-space equation; the state transition matrix for this family is a Laurent polynomial matrix A(σ, σ
^-1
), where σ is the shift operator on R
ⁿ
-valued sequences. We term this family of systems as Laurent systems. We give necessary and sufficient conditions for the exponential ℓ
²
-stability and the exponential ℓ
^∞
-stability of Laurent systems. We also compare the following four different notions of stability for Laurent systems: the ℓ
²
-stability, the exponential ℓ
²
-stability, the ℓ
^∞
-stability, and the exponential ℓ
^∞
-stability; furthermore, we conclude that the ℓ
²
-stability is an outlier.