Impulsive control of nonlocal transport equations

The paper extends an impulsive control-theoretical framework towards dynamical systems in the space of measures. We consider a transport equation describing the time-evolution of a conservative "mass" (probability measure), which represents an infinite ensemble of interacting particles. The driving vector field contains nonlocal terms and is affine in control variable. The control is assumed to be common for all the agents, i.e., it is a function of time variable only. The main feature of the addressed model is the admittance of "shock" impacts, i.e. controls, which can be arbitrary close in their influence on each an agent to Dirac-type distributions. We construct an impulsive relaxation of this system and of the corresponding optimal control problem. For the latter we establish a necessary optimality condition in the form of Pontryagin's Maximum Principle.