The Hautus Test and Genericity Results for Controllable and Uncontrollable Behaviors

The computational effectiveness of Kalman's state space controllability rests on the well-known Hautus test, which describes a rank condition of the matrix $(\frac{d}{dt}I-A, B)$. This paper generalizes this test to a generic class of behaviors (belonging to a Zariski open set) defined by systems of PDE (i.e., systems which arise as kernels of operators given by matrices $(p_{ij}(\partial))$ whose entries are in $\mathbb{C}[\partial_1, \ldots , \partial_n]$) and studies its implications, especially to issues of genericity. The paper distinguishes two classes of systems, underdetermined and overdetermined. The Hautus test developed here implies that a generic strictly underdetermined system is controllable, whereas a generic overdetermined system is uncontrollable. [PUBLICATION ABSTRACT]