Hypocoercivity and hypocontractivity concepts for linear dynamical systems

TL;DR

This paper revisits and extends the concepts of hypocoercivity and hypocontractivity for linear dynamical systems, analyzing their relation to stability and describing short-time behavior through specific indices.

Contribution

It provides a detailed analysis of hypocoercivity and hypocontractivity, linking these concepts to stability, dissipativity, and contractivity, and characterizes short-time propagator behavior using new indices.

Findings

Relations between hypocoercivity/hypocontractivity and stability clarified

Short-time propagator norm behavior characterized by new indices

Extended theoretical framework for continuous and discrete systems

Abstract

For linear dynamical systems (in continuous-time and discrete-time) we revisit and extend the concepts of hypocoercivity and hypocontractivity and give a detailed analysis of the relations of these concepts to (asymptotic) stability, as well as (semi-)dissipativity and (semi-)contractivity, respectively. On the basis of these results, the short-time behavior of the propagator norm for linear continuous-time and discrete-time systems is characterized by the (shifted) hypocoercivity index and the (scaled) hypocontractivity index, respectively.