Isochronous sets of invariant control systems

Let GG be a connected Lie group with Lie algebra gg and Σ=(G,D)Σ=(G,D) a controllable invariant control system. A subset A⊂GA⊂G is said to be isochronous   if there exists a uniform time TA>0TA>0 such that any two arbitrary elements in AA can be connected by a positive orbit of ΣΣ at exact time TATA. In this paper, we search for classes of Lie groups GG such that any ΣΣ has the following property: there exists an increasing sequence of open neighborhoods (Vn)n≥0(Vn)n≥0 of the identity in GG such that the group can be decomposed in isochronous rings Wn=Vn+1−VnWn=Vn+1−Vn. We characterize this property in algebraic terms and we show that three classes of Lie groups satisfy this property: completely solvable simply connected Lie groups, semisimple Lie groups and reductive Lie groups.

► Isochronous sets are sets of controllability in uniform time. ► Invariant control systems on certain Lie groups have annuli of isochronous sets around the identity. ► In particular, completely solvable simply connected Lie groups have this property, as do semisimple and reductive ones.