A decomposition theorem for singular control systems on lie groups

In this paper, we introduce the notion of a singular control system SG on a connected finite-dimensional Lie group G with Lie algebra . This definition depends on a pair of derivations (E,D) of where E plays the same roll as the singular matrix defining S n and D induces the drift vector field of the system. Associated to E we construct a principal fibre bundle and an invariant connection which allow to us to obtain a decomposition result for SG via two subsystems: a linear control system and a differential-algebraic control system. We give an example on the simply connected Heisenberg Lie group of dimension three.