A Note on Impulsive Solutions to Nonlinear Control Systems

In the last decades, many authors provided different notions of impulsive process, seen as a suitably defined limit of a sequence of ordinary processes for a nonlinear control-affine system with unbounded, vector-valued controls. In particular, we refer to the impulsive processes introduced by Karamzin et al. –in which the control is given by a vector measure, a non-negative scalar measure, and a family of so-called attached controls that univocally determine the jumps of the corresponding trajectory– and to the graph completion processes developed by Bressan and Rampazzo et al. –in which an impulsive trajectory is seen as a spatial projection of a Lipschitzian trajectory in space-time. The equivalence between these notions is the crucial assumption of most results on optimal impulsive control problems, such as existence of an optimal process and necessary/sufficient optimality conditions. In this note we exhibit a counterexample which shows that, in presence of state constraints and endpoint constraints involving the total variation of the impulsive control, this equivalence may fail. Thus, we propose to replace the set of impulsive processes with a smaller class of impulsive processes, that we call admissible, which turns out to be actually in one-to-one correspondence with the set of graph completion processes.