Feedback Minimum Principle for Impulsive Processes

We consider an optimal impulsive control problem with a terminal functional and trajectories of bounded variation. The control system we consider has a bilinear structure with respect to the state and control variables and is governed by nonnegative vector Borel measures under constraints on their total variation. This problem is the impulsive-trajectory extension for the corresponding classical optimal control problem, which, in general, does not have optimal solutions with measurable controls. We do not posit any commutativity assumptions guaranteeing the well-posedness property for the impulsive extension. The so-called singular space-time transformation is used to define an individual trajectory and transform the impulsive system to an auxiliary ordinary control system. The aim of this paper is to prove a nonlocal necessary optimality condition for impulsive processes. This condition is based on feedback controls providing descent directions for the functional. This necessary condition is called the feedback minimum principle. It is a generalization of the corresponding principle for classical optimal control problems. The feedback minimum principle is formulated within the framework of the generalized maximum principle for impulsive processes. An example illustrating the optimality condition is considered