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

We consider differential operators A that can be represented by means of a so-called closure relation in terms of a simpler operator \(A_{{\text {ext}}}\) defined on a larger space. We analyse how the spectral properties of A and \(A_{{\text {ext}}}\) are related and give sufficient conditions for exponential stability of the semigroup generated by A in terms of the semigroup generated by \(A_{{\text {ext}}}\). 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.