  We consider constrained partial differential equations of hyperbolic type
with a small parameter $\varepsilon>0$, which turn parabolic in the limit case,
i.e., for $\varepsilon=0$. The well-posedness of the resulting systems is
discussed and the corresponding solutions are compared in terms of the
parameter $\varepsilon$. For the analysis, we consider the system equations as
partial differential-algebraic equation based on the variational formulation of
the problem. For a particular choice of the initial data, we reach first- and
second-order estimates. For general initial data, lower-order estimates are
proven and their optimality is shown numerically.
