A controlled system on a connected Lie group G
(\Sigma) \qquad \dot{g}=\mathcal{X}_g+\sum_{j=1}^m u_jY^j_g
is said to be linear if the drift vector field \mathcal{X} is linear, that is if the flow of \mathcal{X} is a one parameter group of automorphisms, and the Y_j's are right invariant.
Adding a right invariant vector field to \mathcal{X}, one obtains a so-called affine vector field. An affine system is obtained by replacing the drift of a linear system by an affine vector field. Both invariant and linear systems appear as particular cases of affine systems.
The motivation for dealing with such systems is twofold. On the one hand they are natural extensions of invariant systems on Lie groups. On the other one they can be generalized to homogeneous spaces and appear as models for a wide class of systems, on account of the Equivalence Theorem of: [Ph. Jouan, Controllability of linear systems on Lie groups, JDCS, 2011].
Some controllability results will be presented.
Firstly on nilpotent and semisimple Lie groups. In the semisimple case the results are related to time optimal properties of invariant control systems. In the nilpotent one, general controllability conditions are known in the inner derivation case, and on some low dimensional groups.
Second on compact homogeneous spaces, and by virtue of the equivalence theorem, on compact manifolds.