Returning to the Same Point through Bounded Controls in Finite Time

For the Brunovsky system, given an initial point $x^{0}$ in $\mathbb{R}^{2}$, we consider the problem of finding a set of bounded controls that allows to return to the state $x^{0}$ in finite time $T(x^{0})$. We use the Korobov's controllability function method $\Theta(x)$, in particular, the case where $\Theta(x^{0})$ represents the motion time from $x^{0}$ to the same point. We present the solution of the aforementioned problem with the additional condition that the objective is achieved in the optimal time.