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.