  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.
