Geometric analysis of impulse controllability for descriptor system

For descriptor linear system, this paper presents three new concepts: impulsive direction, controllability of impulsive direction, and order of impulsive direction. Controllability at infinity is proved to be equivalent to the controllability of impulsive directions of zero order, which guarantees the controllability of impulsive directions of any order. Unlike what one expects, the controllability of impulsive free-response modes does not guarantee impulse controllability.