Local observability of invariant dynamics on compact Lie groups with square integrable output map functions

In this work, we give a sufficient algebraic condition for the local observability problem of invariant control systems on compact Lie groups such that the output map is not differentiable. In particular, the usual techniques involving Lie derivatives do not work. Our approach comes from the representation theory. We use the regular representation to construct a bilinear system on the Hilbert space of the square integrable function defined on the group to a finite-dimensional vector space. If this bilinear system is observable, then we prove that the invariant control system is locally observable.