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A note on systems with ordinary and impulsive controls

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.