An optimal control problem for the continuity equation is considered. The aim of a "controller" is to maximize the total mass within a target set at a given time moment. The existence of optimal controls is established. For a particular case of the problem, where an initial distribution is absolutely continuous with smooth density and the target set has certain regularity properties, a necessary optimality condition is derived. It is shown that for the general problem one may construct a perturbed problem that satisfies all the assumptions of the necessary optimality condition, and any optimal control for the perturbed problem, is nearly optimal for the original one.2000 Mathematics Subject Classification: 49K20, 49J15

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This paper presents the basic analytical theory for a model that describes interactions between a few individuals (or agents) and a population (a continuum), all moving in R n . The agents affect the population, either repelling or attracting it. Their aim is to steer the population toward a given region K ⊂ R n . This can be seen as a control problem where the state of the system is the set occupied by the population. In this paper we solve simple confinement problems, where the agents' task is to keep the population within a given set. Rigorous analytical results as well as numerical computations are presented.

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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.2000 Mathematics Subject Classification: 49K20, 49J45, 93C20