On the stability of port-Hamiltonian descriptor systems

TL;DR

This paper characterizes the stability of port-Hamiltonian descriptor systems and provides methods to rewrite stable DAEs as dissipative Hamiltonian systems, offering new insights into their structure and stability conditions.

Contribution

It introduces a generalized Lyapunov inequality for stable DAEs and provides conditions to rewrite them as dissipative Hamiltonian systems, advancing the understanding of their stability and structure.

Findings

Stable DAEs can be characterized via a generalized Lyapunov inequality.

Stable DAEs can be rewritten as dissipative Hamiltonian systems on their solution subspace.

Conditions are provided for the stability of dissipative Hamiltonian descriptor systems.

Abstract

We characterize stable differential-algebraic equations (DAEs) using a generalized Lyapunov inequality. The solution of this inequality is then used to rewrite stable DAEs as dissipative Hamiltonian (dH) DAEs on the subspace where the solutions evolve. Conversely, we give sufficient conditions guaranteeing stability of dH DAEs. Further, for stabilizable descriptor systems we construct solutions of generalized algebraic Bernoulli equations which can then be used to rewrite these systems as pH descriptor systems. Furthermore, we show how to describe the stable and stabilizable systems using Dirac and Lagrange structures.