  We investigate an everywhere defined notion of solution for control systems
whose dynamics depend nonlinearly on the control $u$ and state $x,$ and are
affine in the time derivative $\dot u.$ For this reason, the input $u,$ which
is allowed to be Lebesgue integrable, is called impulsive, while a second,
bounded measurable control $v$ is denominated ordinary. The proposed notion of
solution is derived from a topological (non-metric) characterization of a
former concept of solution which was given in the case when the drift is
$v$-independent. Existence, uniqueness and representation of the solution are
studied, and a close analysis of effects of (possibly infinitely many)
discontinuities on a null set is performed as well.
