Abstract
The complete set of solutions to the admissible bounded control problem of the Brunovsky control system of dimension $m$ for $m\geq 2$ via orthogonal polynomials on $[0,T]$ and their second kind polynomials is obtained. For $m\geq 2$ 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 $T$ .
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