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 $$A_{{\text {ext}}}$$ A ext defined on a larger space. We analyse how the spectral properties of A and $$A_{{\text {ext}}}$$ A ext are related and give sufficient conditions for exponential stability of the semigroup generated by A in terms of the semigroup generated by $$A_{{\text {ext}}}$$ A 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.