Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls

In this paper we analyze nonlinear systems whose vector fields are driven by both an ordinary control $v$ and the derivative $\dot u$ of a second control $u$. It is clear that the presence of the derivative $\dot u$ causes some problems in the attempt to define a robust notion of trajectory corresponding to a discontinuous control $u$. On one hand, the nonlinearity of the systems under consideration makes a mere extension of the notion of solution by means of a measure--theoretical approach impossible. On the other hand, the interaction occurring between the derivative $\dot u$ of the {\it impulsive} control $u$ and the ordinary control $v$ prevent us from applying the results of [5], where a space--time extension of the concept of solution was already considered. Indeed in [5] the control $(u,v)$ had equibounded variation, whereas in the present work $v$ is just a measurable bounded map. After giving a stable notion of (space--time) solution to our problem we investigate the topological structure of the set of trajectories: as a consequence of having suppressed the restraint on the variation of $v$ we find that this set is no longer compact. This motivates the search for conditions under which the set of trajectories turns out to be closed again. Finally, in view of several applications we also analyze the case where the derivative $\dot u$ is constrained to range over a closed cone.