On a class of problems of optimal impulse control for a continuity equation

We consider an impulse control problem for a special class of distributed dynamical systems. Such systems result from a relaxation (extension of the set of control processes) of a continuity equation driven by a vector field affine in the control, when there are only integral constraints on the input signals. Problems of this kind appear in the theory of ensemble control and control of multi-agent systems and systems with uncertain initial data. Prior to relaxation, the states of the system may be arbitrarily close to discontinuous curves in the space of probability measures, which leads to the unsolvability of the corresponding extremal problem. The relaxation produces a well-posed optimal control problem for generalized solutions of the continuity equation, which are measure-valued curves with bounded variation. Generalized solutions are described by means of a discontinuous time change in the trajectories of the characteristic system. Some function-theoretic properties of these solutions are studied, and their representation in terms of measure differential equations is obtained. The main result is a necessary optimality condition in the form of the maximum principle for the relaxed problem. Finally, we discuss the possibilities of applying the results for the development of numerical algorithms.