Characterisation for Exponential Stability of port-Hamiltonian Systems

TL;DR

This paper characterizes the exponential stability of energy-dissipating port-Hamiltonian systems by linking energy decay to model ingredients, relaxing regularity conditions on the Hamiltonian density, and providing new stability criteria.

Contribution

It introduces a novel characterization of exponential stability for port-Hamiltonian systems that requires minimal regularity assumptions on the Hamiltonian density.

Findings

Generalized stability criteria applicable to irregular densities.

Established a key assumption involving fundamental solutions of non-autonomous ODEs.

Provided examples of exponentially stable systems with irregular $L_{\infty}$-densities.

Abstract

Given an energy-dissipating port-Hamiltonian system, we characterise the exponential decay of the energy via the model ingredients under mild conditions on the Hamiltonian density $\mathcal{H}$. In passing, we obtain generalisations for sufficient criteria in the literature by making regularity requirements for the Hamiltonian density largely obsolete. The key assumption for the characterisation (and thus the sufficient critera) to work is a uniform bound for a family of fundamental solutions for some non-autonomous, finite-dimensional ODEs. Regularity conditions on $\mathcal{H}$ for previously known criteria such as bounded variation are shown to imply the key assumption. Exponentially stable port-Hamiltonian systems with irregular $L_{\infty}$-densities are provided.