Development of a parameterization method for solving optimal control problems and development of a software package concept

An analysis of existing approaches to the development of software solutions designed to solve optimal control problems is carried out, and a conclusion is drawn about the need to develop specialized numerical software systems. As a numerical method for solving optimal control problems, a parameterization method is proposed, which allows, on the basis of a unified approach, to solve optimal control problems with point or distributed delay and without delay as well. The method describes a scheme for representing a control action in the form of a generalized spline with moving nodes and subsequent reduction of the original optimal control problem with or without delay to a nonlinear programming problem with respect to the spline parameters and temporary nodes. For stated nonlinear programming problem, algorithms for calculating the first and second order derivatives of the objective function are presented. These algorithms make it possible to calculate derivatives based on solving Cauchy problems for direct and adjoint systems. This approach differs from the standard method of calculation based on difference approximation and can significantly reduce the overall amount of calculations. Based on the specifics of the parameterization method, a concept for developing a software package is proposed, and the main provisions of the development are derived. Thus, the software package offers independence in the implementation of methods for solving nonlinear programming problems and discrete schemes for solving Cauchy problems. It also offers a unified (independent of the type of optimal control problem) approach to control parameterization. The results of computational experiments carried out using the parameterization method are also presented. These results confirm the effectiveness of using a unified approach while solving of optimal control problems with point delay, distributed delay, and with no delay.