Abstract
Background. For controlled object which behavior is described by the system of two linear differential equations and with criteria of quality in the form of quadratic functional the optimal control problem is considered. In contrast to the general used methods for investigation of this problem (Pontryagin’s maximum principle, Bellman’s method of dynamic programming) the Lagrange’s multipliers method is proposed by author. Such approach provided the possibility to more effectively obtain the solution of the investigated optimization problem in comparison with the above-mentioned methods.Objective. The purpose of the research is to obtain the formulas for calculation of optimal control and minimal value of the cost functional.Methods. In the paper the methods of calculus of variations were used.Results. In the paper the necessary optimality conditions were obtained and the uniqueness of optimal control was proved. On the basis of these conditions the system of differential Riccati equations is deduced.Conclusions. The solution of this system permits to write the closed formula for optimal control and formula for calculation of minimal value of the performance functional. The results obtained in the paper may be used for investigation of the process of landing of airplane.
Highlights
For controlled object which behavior is described by the system of two linear differential equations and with criteria of quality in the form of quadratic functional the optimal control problem is considered
Для знаходження розв’язку задачі оптимального керування потрібно побудувати математичну модель керованого об’єкта або процесу, що описує його поведінку залежно від часу під впливом керувань
Ключевые слова: квадратичный функционал; метод множителей Лагранжа; необходимые условия оптимальности; оптимальное управление; система дифференциальных уравнений Риккати
Summary
For controlled object which behavior is described by the system of two linear differential equations and with criteria of quality in the form of quadratic functional the optimal control problem is considered. In contrast to the general used methods for investigation of this problem (Pontryagin’s maximum principle, Bellman’s method of dynamic programming) the Lagrange’s multipliers method is proposed by author Such approach provided the possibility to more effectively obtain the solution of the investigated optimization problem in comparison with the above-mentioned methods. The solution of this system permits to write the closed formula for optimal control and formula for calculation of minimal value of the performance functional. Для знаходження розв’язку задачі оптимального керування потрібно побудувати математичну модель керованого об’єкта або процесу, що описує його поведінку залежно від часу під впливом керувань. Математична модель для задачі оптимального керування, як правило, включає в себе систему диференціальних рівнянь, що описують стан керованого об’єкта та формулювання мети керування у вигляді критерію якості керування. Для знаходження розв’язку цієї задачі використовується або принцип максимуму Понтрягіна, або метод динамічного програмування Беллмана
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