On Optimal Harvesting of a Resource on a Circle

This paper studies the optimality in the problem of cyclic harvesting of a resource distributed on a circle with a certain prescribed density. The velocity of motion of the collecting device and the fraction of the resource harvested at a given time play the role of control. The problem is to choose a control maximizing a given quality functional. The paper presents the maximum principle for this (infinite-dimensional) problem. The maximum principle can be written as two inequalities which can be conveniently verified. The class of problems with a concave profit function is solved completely. At the end of the paper, several examples are considered to illustrate the developed technique.