Приближения любого порядка асимптотического решения трехтемповой линейно-квадратичной задачи оптимального управления методом прямой схемы

Abstract

The paper deals with constructing any order asymptotic solution of a class of singularly perturbed linear-quadratic optimal control problems. State variables contain two groups of fast variables. Equations for them have a small parameter or its square before the derivative. The direct scheme method has been used. This method consists of immediate substituting of a postulated asymptotic expansion of a solution into the problem condition and constructing a series of problems, which are simpler than the original one, for finding asymptotics terms. The sought for asymptotic solution contains boundary functions of four types. The direct scheme method allows us to establish the non-increasing of values of the minimized functional if we use a next asymptotic approximation to an optimal control. This fact is illustrated in the paper by an example.