The Stabilization Problem: AGAS and SRS Feedbacks

Abstract

Throughout this paper, M denotes a smooth manifold of dimension n. We are given a control system on M of the form

$\displaystyle \dot x = f ( x, u ) := \sum^m_{i =1} u_i f_i ( x )$,     (1)

where f ₁ , ... , f _m are smooth vector fields on M and where the control

u = (u₁ , ... , u _m )

belongs to the closed unit ball $\overline{B_m}$ in $\rm I\!R^m$. Throughout the paper, “smooth” means “of class $C^\infty$”. We let $x ( \cdot ; \overline x, u ( \cdot ))$ denote the trajectory for (1) for the control u starting at $\bar x \in M$ . Such a control system is said to be globally asymptotically controllable at the point $O \in M$ (abbreviated GAC in the sequel) provided the following two properties are satisfied.