Two energy methods for distributed port-Hamiltonian systems and their application to stability analysis

TL;DR

This paper introduces two local energy methods for analyzing the stability of distributed port-Hamiltonian systems, providing new tools for stability characterization and application to vibrating string networks.

Contribution

It develops novel local energy methods for distributed port-Hamiltonian systems and applies them to stability analysis, including systems with boundary damping and networks of vibrating strings.

Findings

Derived a boundary energy-based exponential stability condition.

Verified the condition on a vibrating string network where previous methods fail.

Analyzed short-time behavior of pH systems with boundary damping.

Abstract

We develop two local energy methods for distributed parameter port-Hamiltonian (pH) systems on one-dimensional spatial domains. The methods are applied to derive a characterization of exponential stability directly in terms of the energy passing through the boundary over a given time horizon. The resulting condition is verified for a network of vibrating strings where existing sufficient conditions cannot be applied. Moreover, we use a local energy method to study the short-time behavior of pH systems with boundary damping which was recently studied in the context of hypocoercivity.