Stability via closure relations with applications to dissipative and port-Hamiltonian systems

TL;DR

This paper introduces a framework using closure relations to analyze the stability of differential operators, with applications to dissipative and port-Hamiltonian systems, providing conditions for exponential stability based on spectral properties.

Contribution

It develops a method to relate spectral properties and stability of operators via closure relations, extending analysis to coupled systems and port-Hamiltonian models.

Findings

Established conditions for exponential stability of differential operators.

Applied the framework to coupled wave-heat and port-Hamiltonian systems.

Analyzed spectral relations between operators and their extensions.

Abstract

We consider differential operators $A$ that can be represented by means of a so-called closure relation in terms of a simpler operator $A_{\operatorname{ext}}$ defined on a larger space. We analyze how the spectral properties of $A$ and $A_{\operatorname{ext}}$ are related and give sufficient conditions for exponential stability of the semigroup generated by $A$ in terms of the semigroup generated by $A_{\operatorname{ext}}$. As applications we study the long-term behaviour of a coupled wave-heat system on an interval, parabolic equations on bounded domains that are coupled by matrix valued potentials, and of linear infinite-dimensional port-Hamiltonian systems with dissipation on an interval.