  Extremum seeking feedback is a powerful method to steer a dynamical system to
an extremum of a partially or completely unknown map. It often requires
advanced system-theoretic tools to understand the qualitative behavior of
extremum seeking systems. In this paper, a novel interpretation of extremum
seeking is introduced. We show that the trajectories of an extremum seeking
system can be approximated by the trajectories of a system which involves
certain Lie brackets of the vector fields of the extremum seeking system. It
turns out that the Lie bracket system directly reveals the optimizing behavior
of the extremum seeking system. Furthermore, we establish a theoretical
foundation and prove that uniform asymptotic stability of the Lie bracket
system implies practical uniform asymptotic stability of the corresponding
extremum seeking system. We use the established results in order to prove local
and semi-global practical uniform asymptotic stability of the extrema of a
certain map for multi-agent extremum seeking systems.

Dynamical Systems