The hypocoercivity index for the short time behavior of linear time-invariant ODE systems

TL;DR

This paper introduces a new algebraic index called the hypocoercivity index to characterize the short-time behavior of conservative-dissipative linear time-invariant ODE systems, linking matrix properties to system dynamics.

Contribution

It provides a concise algebraic characterization of the hypocoercivity index based on the short-time propagator norm, advancing understanding of system stability and behavior.

Findings

Hypocoercivity index characterizes short-time dynamics.

Matrix properties determine system stability.

New algebraic tools for analyzing LTI systems.

Abstract

We consider the class of conservative-dissipative ODE systems, which is a subclass of Lyapunov stable, linear time-invariant ODE systems. We characterize asymptotically stable, conservative-dissipative ODE systems via the hypocoercivity (theory) of their system matrices. Our main result is a concise characterization of the hypocoercivity index (an algebraic structural property of matrices with positive semi-definite Hermitian part introduced in Achleitner, Arnold, and Carlen (2018)) in terms of the short time behavior of the propagator norm for the associated conservative-dissipative ODE system.