  In the present work we investigate topological properties of the set of
controllable differential-algebraic systems of the form
$\tfrac{\text{d}}{\text{d}t}Ex = Ax+Bu$ with real matrices
$E,A\in\mathbb{R}^{\ell\times n}$ and $B\in\mathbb{R}^{\ell\times m}$. We
consider the five controllability concepts free initializability
(controllability at infinity), impulse controllability, controllability in the
behavioural sense, complete controllability and strong controllability. To be
able to make use of the already known algebraic characterizations of these
concepts, we first consider block matrices whose entries are real polynomials
in one indeterminant. We find necessary and sufficient conditions under which
the set of such block matrices, whose rank is "full" in the field of rational
functions or even on the whole complex plane, is generic. Using these results,
we can then for each of the five controllability concepts mentioned above find
necessary and sufficient conditions at $\ell,n$ and $m$, respectively, under
which the set of controllable systems is generic.
