On the optimal control problems constrained by ordinary differential equations using conjugate gradient method and FICO Xpress Mosel

Abstract

In this paper, the general quadratic continuous optimal control problem constrained by an ordinary differential equation is considered. For the analytical solution, the necessary conditions of optimality are applied to the Hamiltonian function. This results in a system of first-order ordinary differential equations that are solved to obtain the optimal state and optimal control variables. In order to obtain the numerical solution, the discretization of the objective function and the corresponding constraints are carried out using 1/3 Simpson’s rule and fifth-order Implicit method respectively. The discretized Optimal Control Problems (OCPs) are converted into unconstrained problems using Augmented Lagrangian Method. The Conjugate Gradient Method (CGM) and Fico Xpress Mosel are used to solve the resulting nonlinear programming problem. Convergence analyses are conducted to determine the effectiveness of the proposed scheme. Two examples are considered to illustrate the robustness of the proposed methods and compare the analysis of the solutions from the CGM and Fico Xpress Mosel. The results show that FICO XPress Mosel performs better than CGM for this class of problems.