Solutions of singular control systems on lie groups

Let G be a connected Lie group with Lie algebra g. A singular control system SG on G is defined by a pair (E;D) of g-derivations. Through a fiber bundle decomposition of TG in [1]; the authors decompose SG in two subsystems SG=V and SV ; as in the linear case on Euclidean spaces, see for instance [9] : Here, V ⊂ G is the Lie subgroup with Lie algebra v; the generalized 0-eigenspace of E: On the other hand, D defines the drift vector field of the system. We assume that the subspace v is invariant under D. With this hypothesis we show a process to determine the solution of SG through every state x = yv; where v is any admissible initial condition on V . From this information, we are able to build the global solution. Finally, in order to illustrate our processes we develop some examples on nilpotent simply connected Lie groups.