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We study convergence and stability properties of control-affine systems. Ourconsiderations are motivated by the problem of stabilizing a control-affinesystem by means of output feedback for states in which the output functionattains an extreme value. Following a recently introduced approach to extremumseeking control, we obtain access to the ascent and descent directions of theoutput function by approximating Lie brackets of suitably defined vectorfields. This approximation is based on convergence properties of trajectoriesof control-affine systems under certain sequences of open-loop controls. Weshow that a suitable notion of convergence for the sequences of open-loopcontrols ensures local uniform convergence of the corresponding sequences offlows. We also show how boundedness properties of the control vector fieldsinfluence the quality of the approximation. Our convergence results aresufficiently strong to derive conditions under which the proposed outputfeedback control law induces exponential stability. We also apply this controllaw to the problem of purely distance-based formation control for nonholonomicmulti-agent systems.

Stabilization of Control-Affine Systems by Local Approximations of Trajectories