Extended set of solutions of a bounded finite-time stabilization problem via the controllability function

For the two-dimensional canonical system, an extended set of bounded finite-time stabilizing positional controls is proposed. For the construction of such controls, which depend on a certain parameter, the Korobov's controllability function method is used. Such an extension is based on the enlarging of the interval of the mentioned parameter, as well as the use of the non-uniqueness of the controllability function for some regions of the phase space ${\mathbb R}^2.$ Additionally, we characterize a region of initial conditions $x^0$ on ${\mathbb R}^2$ of the given system for which the time of motion from $x^0$ to the origin is less than the value of the controllability function.