  In this paper, we describe a broad class of control functions for extremum
seeking problems. We show that it unifies and generalizes existing extremum
seeking strategies which are based on Lie bracket approximations, and allows to
design new controls with favorable properties in extremum seeking and
vibrational stabilization tasks. The second result of this paper is a novel
approach for studying the asymptotic behavior of extremum seeking systems. It
provides a constructive procedure for defining frequencies of control functions
to ensure the practical asymptotic and exponential stability. In contrast to
many known results, we also prove asymptotic and exponential stability in the
sense of Lyapunov for the proposed class of extremum seeking systems under
appropriate assumptions on the vector fields.
