  For a right-invariant control system on a flag manifold $\mathbb{F}_{\Theta}$
of a real semisimple Lie group, we relate the $\mathfrak{a}$-Lyapunov exponents
to the Lyapunov exponents of the system over regular points. Moreover, we adapt
the concept of partial hyperbolicity from the theory of smooth dynamical
systems to control-affine systems, and we completely characterize the partially
hyperbolic chain control sets on $\mathbb{F}_{\Theta}$.
