  This paper introduces extremum seeking (ES) algorithms designed to achieve
perfect tracking of arbitrary time-varying extremum. In contrast to classical
ES approaches that employ constant frequencies and controller gains, our
algorithms leverage time-varying parameters, growing either asymptotically or
exponentially, to achieve desired convergence behaviors. Our stability analysis
involves state transformation, time-dilation transformation, and Lie bracket
averaging. The state transformation is based on the multiplication of the input
state by asymptotic or exponential growth functions. The time transformation
enables tracking of the extremum as it gradually converges to a constant value
when viewed in the dilated time domain. Finally, Lie bracket averaging is
applied to the transformed system, ensuring practical uniform stability in the
dilated time domain as well as asymptotic or exponential stability of the
original system in the original time domain. We validate the feasibility of
these designs through numerical simulations.
