ASYMPTOTİC METHOD SOLUTİON TO AN OPTİMAL MANAGEMENT PROBLEM İN CONTİNUOUS CONDİTİON

Abstract

The problem of optimal control with a variable structure depending on the particular way on the parameter is described in the article. In the considered issue, the object’s motion equations are described by two differential equations depending on the small parameter and these equations are bound to each other. The study investigated the solution of the problem depending on the small parameter that gives a certain functional minimum in the range of solutions described by differential equations, with a small parameter and the necessary conditions for the optimization of the solution are given. The validity of the theorems obtained in this study was proved by applying the Lagrangian principle to the Lagrangian functions and the application of the Farm Theorem to the Lagrangian problem. At the same time, the problem of optimal control with periodic boundary conditions is also explored. It can also be used extensively in solving some practical issues, for example, model in oil extraction by gas—lift method for the case when the reciprocal value of well’s depth represents a small parameter is considered. Problem of optimal mode construction (i.e., construction of optimal program trajectories and controls) is reduced to the linear—quadratic optimal control problem with a small parameter. This issue is investigated using the Lagrange coefficient and the Lagrange principle for the Lagrange problem. Although this issue has been studied in numerous works, the specificity of the issue under consideration makes it possible to investigate it differently. The considered method can be applied to the solution of optimal control problems for the problems described by some nonlinear differential equations.