On the Solution Set of the Admissible Bounded Control Problem via Orthogonal Polynomials

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.