On Chattering Solutions for the Maximum Principle Boundary-Value Problem in the Optimal Control Problem in Microeconomics

Abstract

We consider the problem of optimal control of the maximization of income for the nonlinear mathematical model of microeconomical system used for the description of production, storage, and selling of consumer goods. For this model, we formulate the boundary-value problem of the maximum principle giving the necessary conditions of optimality. We consider conditions for the parameters characterizing the original system under which the solution of the boundary-value problem of maximum principle possesses a section of singular mode. It is shown that this section of singular mode neighbors with nonsingular sections containing containing infinitely many switchings on finite periods of time. For the investigation of this phenomenon, we use the theory developed by M. Zelikin and V. Borisov. It is shown that the solution of the boundary-value problem of maximum principle is locally optimal in the analyzed problem of optimal control. Hence, the corresponding optimal trajectory has three sections. The first part is a nonsingular section in which the original system with infinitely many switchings passes to the singular section for a finite period of time. The second part is the section of singular mode and the last (third) part is a section of the trajectory in which the analyzed system leaves the singular mode with more and more frequent switchings in the inverse time.