The computational effectiveness of Kalman's state space controllability rests on the well-known Hautus test, which describes a rank condition of the matrix (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 derivative)) whose entries are in C[partial derivative(1), ... , partial derivative(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.