Periodic Trajectory Tracking for Control-Affine Driftless Systems on Compact Lie Groups

Abstract

We treat the periodic trajectory tracking problem : given a periodic trajectory of a control-affine, left-invariant driftless system in a compact and connected Lie group G and an initial condition in G , find another trajectory of the system satisfying the initial condition given and that asymptotically tracks the periodic trajectory. We solve this problem locally (for initial conditions in a neighborhood of some point of the periodic trajectory) when G is semisimple and the system is Lie-determined (i.e., controllable), and only for a class of periodic trajectories (which we call regular ). Finally, we present a set of sufficient conditions to ensure the existence of such trajectories.