Impulsive optimal control problems with time delays in the drift term

We introduce a notion of bounded variation solution for a new class of nonlinear control systems with ordinary and impulsive controls, in which the drift function depends not only on the state, but also on its past history, through a finite number of time delays. After proving the well-posedness of such solutions and the continuity of the corresponding input-output map with respect to suitable topologies, we establish necessary optimality conditions for an associated optimal control problem. The approach, which involves approximating the problem by a non-impulsive optimal control problem with time delays and using Ekeland’s principle combined with a recent, nonsmooth version of the Maximum Principle for conventional delayed systems, allows us to deal with mild regularity assumptions and a general endpoint constraint.