  We study the characterization of several distance problems for linear
differential-algebraic systems with dissipative Hamiltonian structure. Since
all models are only approximations of reality and data are always inaccurate,
it is an important question whether a given model is close to a 'bad' model
that could be considered as ill-posed or singular. This is usually done by
computing a distance to the nearest model with such properties. We will discuss
the distance to singularity and the distance to the nearest high index problem
for dissipative Hamiltonian systems. While for general unstructured
differential-algebraic systems the characterization of these distances are
partially open problems, we will show that for dissipative Hamiltonian systems
and related matrix polynomials there exist explicit characterizations that can
be implemented numerically.
