Controllability Function as Time of Motion: An Extension of the Set of Controls in the Two-Dimensional Case

For the two dimensional canonical system, a family of bounded finite-time stabilizing positional controls is extended in terms of a certain parameter. We use Korobov's controllability function, which is a Lyapunov-type function. We focus on the case when the value of the controllability function at the initial position is exactly the time of motion from the initial point to the origin. As one of the consequences of such an extension, we ascertain that to each bounded positional control there could correspond one, two or three different times of motion from a given position to the origin.