The complete set of solutions to the admissible bounded control problem of the Brunovsky control system of dimension m for m >= 2 via orthogonal polynomials on [0, T] and their second kind polynomials is obtained. For m >= 2 given an initial position x(0) and a time T greater than the optimal time t(min), we prove that there are only two bang-bang controls with m switchings that steer the state trajectory to the origin exactly at time T.