A note on systems with ordinary and impulsive controls

Abstract

We investigate an everywhere defined notion of solution for control systems whose dynamics depend non-linearly on the control u and state x, and are affine in the time derivative 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 (nonmetric) 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.