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 <inline-formula><tex-math notation="LaTeX">$A(\sigma,\sigma ^{-1})$</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">$\sigma$</tex-math></inline-formula> is the shift operator on <inline-formula><tex-math notation="LaTeX">$\mathbb {R}^n$</tex-math></inline-formula>-valued sequences. We term this family of systems as Laurent systems. We give necessary and sufficient conditions for the exponential <inline-formula><tex-math notation="LaTeX">$\ell ^2$</tex-math></inline-formula>-stability and the exponential <inline-formula><tex-math notation="LaTeX">$\ell ^{\infty }$</tex-math></inline-formula>-stability of Laurent systems. We also compare the following four different notions of stability for Laurent systems: the <inline-formula><tex-math notation="LaTeX">$\ell ^2$</tex-math></inline-formula>-stability, the exponential <inline-formula><tex-math notation="LaTeX">$\ell ^2$</tex-math></inline-formula>-stability, the <inline-formula><tex-math notation="LaTeX">$\ell ^{\infty }$</tex-math></inline-formula>-stability, and the exponential <inline-formula><tex-math notation="LaTeX">$\ell ^{\infty }$</tex-math></inline-formula>-stability; furthermore, we conclude that the <inline-formula><tex-math notation="LaTeX">$\ell ^2$</tex-math></inline-formula>-stability is an outlier.