18th International Geometry Symposium, Malatya, Turkey, 12 - 13 July 2021, pp.45
A control system on a connected Lie group
\begin{equation}\label{control}
\dot{q} = \mathcal{X}_{q}+\sum_{i=1}^{n} u_i Y_{q}^{i}
\end{equation}
is called to be linear if drift vector field $\mathcal{X}$ is linear which means that the flow is a 1-parameter group of automorphisms and $Y^{i}$'s are right invariant vector fields. When a right invariant vector field is added to drift vector field $\mathcal{X}$, the vector field obtained is called affine. The system (\ref{control}) defined on a manifold is equivalent to a linear system or a homogeneous space under some conditions via diffeomorphism \cite{Jouan}. Therefore, it becomes important under which conditions the vector fields on such systems have projections on a homogeneous space.\\
In this paper, by using differential geometric and Lie theoretic approach, we mention projections of vector fields and then apply them to the control system.