線形動的システムのリーチャビリテイ問題

最近dx/dt = Ax+Bu の形かある数式に関して学んでいます。
One issue of confusion was the continuous time equivalent of the reachable set. Basically, it was {Cu}, where C is the matrix
[B AB A^2B ... A^(n-1)B]. However, in order to derive the minimal input norm to reach a desired state, discretization was carried out. Thus, Ad and Bd were defined, with Ad = e^(Ah) and Bd = integral(0,h)(e^At)dtB. I am not sure how these two formulas were derived.
In addition, a proof that any vector in the range of the controllability matrix C was reachable was given. This was based on the use of an impulse input comprised of the kth derivatives of the delta function. I am not sure why this function was useful, but the proof seemed to indicate that x(0+) could be written as a product of C and f. Apparently, x(0+) represented a reachable point which is in the range of the controllability matrix. Given all possible f vectors, apparently the entire range of C would be covered and different reachable points obtained.