A decomposition theorem for singular control systems on lie groups

In this paper, we introduce the notion of a singular control system S_G on a connected finite-dimensional Lie group G with Lie algebra g. This definition depends on a pair of derivations (E, D) of g where E plays the same roll as the singular matrix defining S_ℝⁿ 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 S_G 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.