Returning to the Same Point through Bounded Controls in Finite Time

Abstract

For the Brunovsky system, given an initial point <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$x^{0}$</tex> in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{R}^{2}$</tex>, we consider the problem of finding a set of bounded controls that allows to return to the state <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$x^{0}$</tex> in finite time <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$T(x^{0})$</tex>. We use the Korobov's controllability function method <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Theta(x)$</tex>, in particular, the case where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Theta(x^{0})$</tex> represents the motion time from <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$x^{0}$</tex> 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.