Solutions of singular control systems on lie groups

Abstract

Let G be a connected Lie group with Lie algebra $$ \mathfrak{g} $$ . A singular control system $$ {\mathcal{S}_G} $$ on G is defined by a pair (E, D) of $$ \mathfrak{g} $$ -derivations. Through a fiber bundle decomposition of TG in [1] ; the authors decompose $$ {\mathcal{S}_G} $$ in two subsystems $$ {\mathcal{S}_{G/V}} $$ and $$ {\mathcal{S}_V} $$ ; as in the linear case on Euclidean spaces, see for instance [9] : Here, $$ V \subset G $$ is the Lie subgroup with Lie algebra $$ \mathfrak{v} $$ ; the generalized 0-eigenspace of E: On the other hand, D defines the drift vector field of the system. We assume that the subspace $$ \mathfrak{v} $$ is invariant under D. With this hypothesis we show a process to determine the solution of $$ {\mathcal{S}_G} $$ 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.