Minimal set of generators of controllability space for singular linear systems

In recent years, there has been increasing the interest in the descriptive analysis of singular (also called generalized) systems in the form $E \dot x(t) = Ax(t)$ because they play important roles in mathematical modelling problems permeating many aspects of daily life arising in a wide range of applications. Considerable advances have been obtained in the description of their structural and dynamical properties. However, much less effort has been devoted to studying the exact controllability measuring the minimum set of controls that are needed to steer the whole system $E \dot x(t) = Ax(t)$ toward any desired state. In this paper, we focus the study on the obtention of the set of all $B$ making the system $E \dot x(t) = Ax(t)+Bu(t)$ controllable