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.

中文翻译：

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紧凑李群上仿射无漂移系统的周期性轨迹跟踪

我们对待周期性轨迹跟踪问题：给定一个控制仿射的周期轨迹，左不变无碛系统在一个紧凑和连接李群摹和初始条件摹，发现系统的另一个轨迹满足给定的初始条件和渐近地跟踪周期轨迹。当G是半简单的并且系统是李确定的（即可控制的）时，我们仅针对一类周期性轨迹（我们称其为正则）局部解决此问题（对于周期性轨迹某个点附近的初始条件））。最后，我们提出了一组足够的条件来确保这种轨迹的存在。