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

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.