This paper concerns the structure that can be achieved by feedback in descriptor systems that lack controllability at infinity. Staircase and double staircase condensed forms obtained through a sequence of orthogonal state transformations display when and how feedback can be used to achieve minimal index. Furthermore, they reveal that the modes that are uncontrollable at infinity have a fixed minimal index that cannot be reduced by feedback. However, this fixed higher index part of the control system is constrained to be zero in an appropriate coordinate system, provided the initial conditions are consistent. The remainder is a reduced order system that is controllable at infinity that can be made to have index one by feedback.