Linear differential-algebraic systems are generically controllable

Abstract

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.