  We study convergence and stability properties of control-affine systems. Our
considerations are motivated by the problem of stabilizing a control-affine
system by means of output feedback for states in which the output function
attains an extreme value. Following a recently introduced approach to extremum
seeking control, we obtain access to the ascent and descent directions of the
output function by approximating Lie brackets of suitably defined vector
fields. This approximation is based on convergence properties of trajectories
of control-affine systems under certain sequences of open-loop controls. We
show that a suitable notion of convergence for the sequences of open-loop
controls ensures local uniform convergence of the corresponding sequences of
flows. We also show how boundedness properties of the control vector fields
influence the quality of the approximation. Our convergence results are
sufficiently strong to derive conditions under which the proposed output
feedback control law induces exponential stability. We also apply this control
law to the problem of purely distance-based formation control for nonholonomic
multi-agent systems.
