# Periodic trajectory tracking for control-affine driftless systems on compact Lie groups

## TL;DR

This paper addresses the problem of asymptotic periodic trajectory tracking for control-affine, driftless systems on compact Lie groups, providing local solutions under specific controllability and regularity conditions.

## Contribution

It introduces a local solution to the periodic trajectory tracking problem on compact Lie groups for semisimple cases with controllability, and establishes conditions for trajectory existence.

## Key findings

- Solution exists locally for semisimple Lie groups and controllable systems.
- Provides sufficient conditions for the existence of periodic trajectories.
- Addresses tracking for a class of regular periodic trajectories.

## 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.